6 files changed, 17842 insertions, 9390 deletions

diff --git a/af/n.lyx b/af/n.lyx
index fddb51e..889cf77 100644
--- a/af/n.lyx
+++ b/af/n.lyx
@@ -135,7 +135,40 @@ filename "../license.lyx"
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+urldef
+\backslash
+sigmafinite
+\backslash
+url{https://en.wikipedia.org/wiki/%CE%A3-finite_measure}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Bibliografía:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Itemize
@@ -295,12 +328,14 @@ Wikipedia, the Free Encyclopedia.
 
 \lang spanish
  Recuperado de 
-\begin_inset Flex URL
+\begin_inset ERT
 status open
 
 \begin_layout Plain Layout
 
-https://en.wikipedia.org/wiki/%CE%A3-finite_measure
+
+\backslash
+sigmafinite{}
 \end_layout
 
 \end_inset
@@ -308,6 +343,46 @@ https://en.wikipedia.org/wiki/%CE%A3-finite_measure
  el 13 de enero de 2023.
 \end_layout
 
+\begin_layout Itemize
+
+\lang english
+Wikipedia, the Free Encyclopedia.
+ 
+\emph on
+Tychonoff space.
+
+\emph default
+\lang spanish
+ Recuperado de 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+https://en.wikipedia.org/wiki/Tychonoff_space
+\end_layout
+
+\end_inset
+
+ el 17 de enero de 2023.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Chapter
 Espacios de Banach
 \end_layout
@@ -329,7 +404,7 @@ Espacios de Hilbert
 \begin_layout Standard
 \begin_inset CommandInset include
 LatexCommand input
-filename "n1b.lyx"
+filename "n2.lyx"
 
 \end_inset
 
@@ -343,7 +418,21 @@ Teoría espectral
 \begin_layout Standard
 \begin_inset CommandInset include
 LatexCommand input
-filename "n2.lyx"
+filename "n3.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Principios fundamentales
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n4.lyx"
 
 \end_inset
 
diff --git a/af/n1.lyx b/af/n1.lyx
index fe32176..ee3a964 100644
--- a/af/n1.lyx
+++ b/af/n1.lyx
@@ -120,411 +120,1273 @@ lineal conjugada
 \end_layout
 
 \begin_layout Section
-Espacios de Banach
+Espacios vectoriales topológicos
 \end_layout
 
 \begin_layout Standard
-Dados un 
+Un 
+\series bold
+espacio vectorial topológico
+\series default
+ (
+\series bold
+e.v.t.
+\series default
+) es un espacio topológico 
+\begin_inset Formula $(E,{\cal T})$
+\end_inset
+
+ donde 
+\begin_inset Formula $E$
+\end_inset
+
+ es un 
 \begin_inset Formula $\mathbb{K}$
 \end_inset
 
--espacio vectorial 
-\begin_inset Formula $X$
+-espacio vectorial y 
+\begin_inset Formula $s:E\times E\to E$
 \end_inset
 
  y 
-\begin_inset Formula $A\subseteq X$
+\begin_inset Formula $p:\mathbb{K}\times E\to E$
 \end_inset
 
-, llamamos 
-\begin_inset Formula $\text{span}A$
+ dadas por 
+\begin_inset Formula $s(x,y)\coloneqq x+y$
 \end_inset
 
- al menor subespacio vectorial de 
-\begin_inset Formula $X$
+ y 
+\begin_inset Formula $p(\alpha,x)\coloneqq\alpha x$
 \end_inset
 
- que contiene a 
-\begin_inset Formula $A$
+ son continuas en la topología producto, y entonces 
+\begin_inset Formula ${\cal T}$
 \end_inset
 
-, y decimos que una 
-\begin_inset Formula $q:X\to\mathbb{R}$
+ es una 
+\series bold
+topología vectorial
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $\mathbb{K}$
 \end_inset
 
- es:
-\end_layout
+-espacios vectoriales 
+\begin_inset Formula $E$
+\end_inset
 
-\begin_layout Enumerate
+ y 
+\begin_inset Formula $F$
+\end_inset
 
+, un 
 \series bold
-Subaditiva
+operador
 \series default
- si 
-\begin_inset Formula $\forall x,y\in X,q(x+y)\leq q(x)+q(y)$
+ es una función lineal de 
+\begin_inset Formula $E$
 \end_inset
 
-.
-\end_layout
+ a 
+\begin_inset Formula $F$
+\end_inset
 
-\begin_layout Enumerate
+, y llamamos 
+\series bold
+dual algebraico
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
 
+ al conjunto de funciones de 
+\begin_inset Formula $E$
+\end_inset
+
+ a 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+, llamadas 
 \series bold
-Positivamente homogénea
+formas lineales
 \series default
- si 
-\begin_inset Formula $\forall a\in\mathbb{K}\cap\mathbb{R}^{+},\forall x\in X,q(ax)=aq(x)$
+ de 
+\begin_inset Formula $E$
 \end_inset
 
 .
-\end_layout
+ Si 
+\begin_inset Formula $E$
+\end_inset
 
-\begin_layout Enumerate
+ y 
+\begin_inset Formula $F$
+\end_inset
+
+ son 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-e.v.t.s, 
+\begin_inset Formula ${\cal L}(E,F)$
+\end_inset
+
+ es el conjunto de operadores continuos de 
+\begin_inset Formula $E$
+\end_inset
+
+ a 
+\begin_inset Formula $F$
+\end_inset
 
+, y llamamos 
 \series bold
-Absolutamente homogénea
+dual topológico
 \series default
- si 
-\begin_inset Formula $\forall a\in\mathbb{K},\forall x\in X,q(ax)=|a|q(x)$
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+ a 
+\begin_inset Formula $E'\coloneqq{\cal L}(E,\mathbb{K})$
 \end_inset
 
 .
 \end_layout
 
-\begin_layout Enumerate
-Una 
-\series bold
-seminorma
-\series default
- si es subaditiva y absolutamente homogénea.
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{TEM}
 \end_layout
 
-\begin_layout Enumerate
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Una 
 \series bold
-norma
+base de entornos
 \series default
- si es una seminorma con 
-\begin_inset Formula $q^{-1}(0)=0$
+ de 
+\begin_inset Formula $p\in X$
+\end_inset
+
+ es una subfamilia 
+\begin_inset Formula ${\cal B}(p)\subseteq{\cal E}(p)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall V\in{\cal E}(p),\exists U\in{\cal B}(p):U\subseteq V$
 \end_inset
 
 .
+ [...] Un espacio topológico [...] satisface el 
+\series bold
+primer axioma de numerabilidad
+\series default
+, o es 
+\series bold
+1AN
+\series default
+, si todo punto posee una base de entornos numerable [...].
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
-Toda norma es definida positiva
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-e.v.t.:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $s_{a}:E\to E$
+\end_inset
+
+ con 
+\begin_inset Formula $y\in E$
+\end_inset
+
+ y 
+\begin_inset Formula $p_{\lambda}:E\to E$
+\end_inset
+
+ con 
+\begin_inset Formula $\lambda\in\mathbb{K}^{*}$
+\end_inset
+
+ dados por 
+\begin_inset Formula $s_{a}(x)\coloneqq x+a$
+\end_inset
+
+ y 
+\begin_inset Formula $p_{\lambda}(x)\coloneqq\lambda x$
+\end_inset
+
+ son homeomorfismos.
 \begin_inset Note Comment
 status open
 
 \begin_layout Plain Layout
-, pues si 
-\begin_inset Formula $x\in X\setminus0$
+\begin_inset Formula $s_{a}$
 \end_inset
 
-, 
-\begin_inset Formula $q(x)=|-1|q(x)=q(-x)\neq0$
+ es la composición de 
+\begin_inset Formula $x\mapsto(x,a)$
 \end_inset
 
-, pero 
-\begin_inset Formula $0=q(0)=q(x-x)\leq q(x)+q(-x)=2q(x)$
+ con la suma, por lo que es continua, y análogamente lo es 
+\begin_inset Formula $p_{\lambda}$
+\end_inset
+
+, pero la inversa de 
+\begin_inset Formula $s_{a}$
+\end_inset
+
+ es 
+\begin_inset Formula $s_{-a}$
+\end_inset
+
+ y la de 
+\begin_inset Formula $p_{\lambda}$
+\end_inset
+
+ es 
+\begin_inset Formula $p_{\lambda^{-1}}$
+\end_inset
+
+, que también son continuas.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+La suma de un abierto y un subconjunto cualquiera de 
+\begin_inset Formula $E$
+\end_inset
+
+ es abierta en 
+\begin_inset Formula $E$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sean 
+\begin_inset Formula $G\subseteq E$
+\end_inset
+
+ abierto y 
+\begin_inset Formula $A\subseteq E$
+\end_inset
+
+.
+ Todo 
+\begin_inset Formula $p\in G+A$
+\end_inset
+
+ es de la forma 
+\begin_inset Formula $p=g+a$
+\end_inset
+
+ con 
+\begin_inset Formula $g\in G$
 \end_inset
 
  y 
-\begin_inset Formula $q(x)>0$
+\begin_inset Formula $a\in A$
+\end_inset
+
+, pero entonces 
+\begin_inset Formula $G+a\subseteq G+A$
+\end_inset
+
+ es un entorno de 
+\begin_inset Formula $g+a$
+\end_inset
+
+ por el homeomorfismo 
+\begin_inset Formula $s_{a}$
+\end_inset
+
+.
+\end_layout
+
 \end_inset
 
 
 \end_layout
 
+\begin_layout Enumerate
+La suma de un cerrado y un compacto de 
+\begin_inset Formula $E$
+\end_inset
+
+ es cerrada en 
+\begin_inset Formula $E$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sean 
+\begin_inset Formula $F$
+\end_inset
+
+ el cerrado y 
+\begin_inset Formula $K$
+\end_inset
+
+ el compacto, tomamos una sucesión convergente arbitraria en 
+\begin_inset Formula $F+K$
+\end_inset
+
+, 
+\begin_inset Formula $(x_{n}+y_{n})_{n}$
+\end_inset
+
+ con cada 
+\begin_inset Formula $x_{n}\in F$
+\end_inset
+
+ y cada 
+\begin_inset Formula $y_{n}\in K$
+\end_inset
+
+, y 
+\begin_inset Formula $z\coloneqq\lim_{n}(x_{n}+y_{n})$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $K$
+\end_inset
+
+ es compacto, existe una subsucesión 
+\begin_inset Formula $(x_{n_{k}})_{k}$
+\end_inset
+
+ convergente a un 
+\begin_inset Formula $x\in K$
+\end_inset
+
+, luego 
+\begin_inset Formula $(y_{n_{k}})_{k}$
+\end_inset
+
+ converge a 
+\begin_inset Formula $z-x\in F$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $z=(z-x)+x\in F+K$
 \end_inset
 
 .
 \end_layout
 
-\begin_layout Standard
-Un 
-\series bold
-espacio normado
-\series default
- es un 
-\begin_inset Formula $\mathbb{K}$
 \end_inset
 
--espacio vectorial 
+
+\end_layout
+
+\begin_layout Enumerate
+Un subespacio vectorial de 
 \begin_inset Formula $X$
 \end_inset
 
- con una norma 
-\begin_inset Formula $\Vert\cdot\Vert:X\to\mathbb{R}$
+ es propio si y sólo si su interior es vacío.
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $Y<X$
+\end_inset
+
+ un subespacio vectorial propio y 
+\begin_inset Formula $p\in X\setminus Y$
+\end_inset
+
+, para 
+\begin_inset Formula $y\in Y$
+\end_inset
+
+, por continuidad de la suma y el producto, 
+\begin_inset Formula $\lim_{h\to0}(y+hp)=y$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $(y+\frac{p}{n})_{n\in\mathbb{N}^{*}}$
+\end_inset
+
+ es una sucesión de elementos de 
+\begin_inset Formula $X\setminus Y$
+\end_inset
+
+ que converge a 
+\begin_inset Formula $y$
+\end_inset
+
+, 
+\begin_inset Formula $y\notin\mathring{Y}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathring{Y}=\emptyset$
 \end_inset
 
 .
- Todo espacio normado 
-\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
 \end_inset
 
- es un espacio métrico con la distancia 
-\begin_inset Formula $(x,y)\mapsto\Vert x-y\Vert$
+
+\end_layout
+
 \end_inset
 
-, y llamamos 
-\begin_inset Formula $B_{X}\coloneqq B[0,1]=\overline{B(0,1)}=\{x\in X\mid\Vert x\Vert\leq1\}$
+El contrarrecíproco es trivial.
+\end_layout
+
 \end_inset
 
- y conjunto de 
-\series bold
-vectores unitarios
-\series default
- a 
-\begin_inset Formula $S_{X}\coloneqq\partial B(0,1)=\{x\in X\mid\Vert x\Vert=1\}$
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $F\subseteq E$
+\end_inset
+
+ es un subespacio vectorial también lo es 
+\begin_inset Formula $\overline{F}$
 \end_inset
 
 .
- La norma es uniformemente continua en este espacio métrico
 \begin_inset Note Comment
 status open
 
 \begin_layout Plain Layout
-, pues para 
-\begin_inset Formula $\varepsilon>0$
+Dados 
+\begin_inset Formula $a\in\mathbb{K}$
 \end_inset
 
-, si 
-\begin_inset Formula $x,y\in X$
+ y 
+\begin_inset Formula $x,y\in\overline{F}$
 \end_inset
 
- cumplen 
-\begin_inset Formula $\Vert x-y\Vert<\varepsilon$
+, 
+\begin_inset Formula $x$
 \end_inset
 
-, por subaditividad es 
-\begin_inset Formula $\Vert x\Vert\leq\Vert x-y\Vert+\Vert y\Vert$
+ e 
+\begin_inset Formula $y$
 \end_inset
 
- y por tanto 
-\begin_inset Formula $\left|\Vert x\Vert-\Vert y\Vert\right|=\Vert x\Vert-\Vert y\Vert\leq\Vert x-y\Vert<\varepsilon$
+ son límites de sucesiones respectivas 
+\begin_inset Formula $\{x_{n}\}_{n},\{y_{n}\}_{n}\subseteq F$
 \end_inset
 
+, con lo que 
+\begin_inset Formula $x+y=\lim_{n}(x_{n}+y_{n})\in\overline{F}$
+\end_inset
 
+ y 
+\begin_inset Formula $ax=\lim_{n}ax_{n}\in\overline{F}$
+\end_inset
+
+.
 \end_layout
 
 \end_inset
 
-.
- Un vector es 
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $F$
+\end_inset
+
+ es otro e.v.t.
+ y 
+\begin_inset Formula $T:E\to F$
+\end_inset
+
+ es lineal, 
+\begin_inset Formula $T$
+\end_inset
+
+ es continua si y sólo si lo es en 0, y si 
+\begin_inset Formula $F=\mathbb{K}$
+\end_inset
+
+ con la topología usual, 
+\begin_inset Formula $T$
+\end_inset
+
+ es continua si y sólo si 
+\begin_inset Formula $\ker T\leq E$
+\end_inset
+
+ es cerrado.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A\subseteq E$
+\end_inset
+
+ es 
 \series bold
-unitario
+equilibrado
 \series default
- si tiene norma 1.
- Un 
+ si 
+\begin_inset Formula $\forall\alpha\in\mathbb{K},(|\alpha|\leq1\implies\alpha A\subseteq A)$
+\end_inset
+
+, es 
 \series bold
-espacio de Banach
+absorbente
 \series default
- es un espacio normado completo.
+ si 
+\begin_inset Formula $\forall x\in E,\exists\rho_{0}>0:\forall\rho\in\mathbb{K},(|\rho|\geq\rho_{0}\implies x\in\rho A)$
+\end_inset
+
+, y es 
+\series bold
+total
+\series default
+ si 
+\begin_inset Formula $\overline{\text{span}A}=E$
+\end_inset
+
+.
+ Los entornos de 0 son absorbentes.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
-\begin_inset ERT
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-e.v.t.
+ y 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+ una base de entornos de 0:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $x\in E$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in\mathbb{K}^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $x+\alpha{\cal U}$
+\end_inset
+
+ es base de entornos de 
+\begin_inset Formula $x$
+\end_inset
+
+.
+\begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
 
 
-\backslash
-begin{samepage}
 \end_layout
 
+\begin_layout Enumerate
+\begin_inset Formula $\forall M\subseteq E,\overline{M}=\bigcap_{U\in{\cal U}}(M+U)$
 \end_inset
 
+.
+\begin_inset Note Note
+status open
 
+\begin_layout Plain Layout
+nproof
 \end_layout
 
-\begin_layout Standard
-Sea 
-\begin_inset Formula $(X,\Vert\cdot\Vert)$
 \end_inset
 
- un 
-\begin_inset Formula $\mathbb{K}$
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall U\in{\cal U},\exists V\in{\cal U}:V+V\subseteq U$
 \end_inset
 
--espacio normado:
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Enumerate
-Todo subespacio vectorial de 
-\begin_inset Formula $X$
+\begin_inset Formula $\forall U\in{\cal U},\exists V\in{\cal U}:\forall\alpha\in\mathbb{K},(|\alpha|\leq1\implies\alpha V\subseteq U)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
- es normado con la norma inducida.
+
 \end_layout
 
 \begin_layout Enumerate
-\begin_inset Formula $s:X\times X\to X$
+Todo 
+\begin_inset Formula $U\in{\cal U}$
+\end_inset
+
+ es absorbente.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\tilde{{\cal U}}\coloneqq\left\{ \bigcup_{|\alpha|\leq1}\alpha U\right\} _{U\in{\cal U}}$
 \end_inset
 
  y 
-\begin_inset Formula $p:\mathbb{K}\times X\to X$
+\begin_inset Formula $\overline{{\cal U}}\coloneqq\{\overline{U}\}_{U\in{\cal U}}$
 \end_inset
 
- dadas por 
-\begin_inset Formula $s(x,y)\coloneqq x+y$
+ son bases de entornos de 0, con lo que toda e.v.t.
+ tiene una base de entornos del 0 formada por conjuntos absorbentes, equilibrado
+s y cerrados.
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+base de filtro
+\series default
+ en un conjunto 
+\begin_inset Formula $S$
+\end_inset
+
+ es un 
+\begin_inset Formula ${\cal U}\subseteq{\cal P}(S)$
+\end_inset
+
+ no vacío tal que 
+\begin_inset Formula $\forall U,V\in{\cal U},\exists W\in{\cal U}:W\subseteq U\cap V$
+\end_inset
+
+, y se puede definir una topología en 
+\begin_inset Formula $S$
+\end_inset
+
+ tomando una base de filtros sobre cada punto, que actuará como base de
+ entornos.
+\end_layout
+
+\begin_layout Section
+Espacios localmente convexos
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $E$
+\end_inset
+
+ un espacio vectorial y 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+ una base de filtro en 
+\begin_inset Formula $E$
+\end_inset
+
+ formada por conjuntos absorbentes y equilibrados y tal que 
+\begin_inset Formula $\bigcap{\cal U}=0$
 \end_inset
 
  y 
-\begin_inset Formula $p(a,x)\coloneqq ax$
+\begin_inset Formula $\forall U\in{\cal U},\exists V\in{\cal U}:V+V\subseteq U$
 \end_inset
 
- son continuas.
-\begin_inset Note Comment
+, existe una única topología vectorial sobre 
+\begin_inset Formula $E$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $x\in E$
+\end_inset
+
+, 
+\begin_inset Formula $\{x+U\}_{U\in{\cal U}}$
+\end_inset
+
+ es base de entornos de 
+\begin_inset Formula $x$
+\end_inset
+
+.
+\begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-Sea 
-\begin_inset Formula $A\subseteq X$
+nproof
+\end_layout
+
 \end_inset
 
- abierto, queremos ver que 
-\begin_inset Formula $s^{-1}(A)$
+
+\end_layout
+
+\begin_layout Standard
+Dado un 
+\begin_inset Formula $\mathbb{K}$
 \end_inset
 
- y 
-\begin_inset Formula $p^{-1}(A)$
+-espacio vectorial 
+\begin_inset Formula $E$
 \end_inset
 
- son abiertos con la topología producto.
- Sean 
-\begin_inset Formula $(x,y)\in s^{-1}(A)$
+, 
+\begin_inset Formula $q:E\to\mathbb{R}$
+\end_inset
+
+ es 
+\series bold
+subaditiva
+\series default
+ si 
+\begin_inset Formula $\forall x,y\in E,q(x+y)\leq q(x)+q(y)$
+\end_inset
+
+, 
+\series bold
+positivamente homogénea
+\series default
+ si 
+\begin_inset Formula $\forall\lambda\in\mathbb{R}^{+},\forall x\in E,q(\lambda x)=\lambda q(x)$
 \end_inset
 
  y 
-\begin_inset Formula $b\coloneqq s(x,y)$
+\series bold
+absolutamente homogénea
+\series default
+ si 
+\begin_inset Formula $\forall\lambda\in\mathbb{K},\forall x\in E,q(\lambda x)=|\lambda|q(x)$
 \end_inset
 
-, existe 
-\begin_inset Formula $\varepsilon>0$
+.
+ Una 
+\series bold
+seminorma
+\series default
+ es una función 
+\begin_inset Formula $E\to\mathbb{R}$
 \end_inset
 
- tal que 
-\begin_inset Formula $B(b,\varepsilon)\subseteq A$
+ subaditiva y absolutamente homogénea.
+ Las seminormas son no negativas
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues si 
+\begin_inset Formula $q:E\to\mathbb{R}$
 \end_inset
 
-, pero entonces, para 
-\begin_inset Formula $(x',y')\in B(x,\frac{\varepsilon}{2})\times B(y,\frac{\varepsilon}{2})$
+ es una seminorma y 
+\begin_inset Formula $x\in X$
 \end_inset
 
 , 
+\begin_inset Formula $0=0q(x)=q(0x)=q(0)=q(x-x)\leq q(x)+q(-x)=q(x)+|-1|q(x)=2q(x)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $E$
+\end_inset
+
+ un espacio vectorial y 
+\begin_inset Formula ${\cal P}\subseteq\mathbb{R}^{E}$
+\end_inset
+
+ una familia de seminormas con 
+\begin_inset Formula $\bigcap_{p\in{\cal P}}\{x\in E\mid p(x)=0\}=0$
+\end_inset
+
+,
 \begin_inset Formula 
 \[
-\Vert s(x',y')-b\Vert=\Vert\cancel{b}+(x'-x)+(y'-y)\cancel{-b}\Vert\leq\Vert x'-x\Vert+\Vert y'-y\Vert<\varepsilon,
+{\cal U}\coloneqq\left\{ \bigcap_{p\in{\cal F}}\{x\in E\mid p(x)<\varepsilon\}\right\} _{{\cal F}\subseteq{\cal P}\text{ finito},\varepsilon>0}
 \]
 
 \end_inset
 
-luego 
-\begin_inset Formula $s(x',y')\in B(x,\frac{\varepsilon}{2})\subseteq A$
+es una base de filtro formada por conjuntos convexos, absorbentes y equilibrados
+, con intersección vacía y tal que para 
+\begin_inset Formula $U\in{\cal U}$
+\end_inset
+
+ existe 
+\begin_inset Formula $V\in{\cal V}$
+\end_inset
+
+ con 
+\begin_inset Formula $V+V\subseteq U$
+\end_inset
+
+, y llamamos 
+\series bold
+topología asociada a 
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+
+\series default
+ a la única topología vectorial sobre 
+\begin_inset Formula $E$
+\end_inset
+
+ que tiene a 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+ como base de entornos de 0.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio vectorial, 
+\begin_inset Formula $A\subseteq E$
+\end_inset
+
+ es 
+\series bold
+absolutamente convexo
+\series default
+ si es convexo y equilibrado, si y sólo si 
+\begin_inset Formula $\forall x,y\in A,\forall\alpha,\beta\in\mathbb{K},(|\alpha|+|\beta|\leq1\implies\alpha x+\beta y\in A)$
 \end_inset
 
 .
- Sean 
-\begin_inset Formula $(a,x)\subseteq p^{-1}(A)$
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
- y 
-\begin_inset Formula $b\coloneqq p(a,x)$
+
+\end_layout
+
+\begin_layout Standard
+La intersección de conjuntos absolutamente convexos es absolutamente convexa,
+ y llamamos 
+\series bold
+envoltura absolutamente convexa
+\series default
+ de 
+\begin_inset Formula $A\subseteq E$
 \end_inset
 
-, existe 
-\begin_inset Formula $\varepsilon\in(0,1)$
+, 
+\begin_inset Formula $\Gamma(A)$
 \end_inset
 
- tal que 
-\begin_inset Formula $B(b,\varepsilon)\subseteq A$
+ a la intersección de todos los conjuntos absolutamente convexos que contienen
+ a 
+\begin_inset Formula $A$
 \end_inset
 
-, pero entonces para 
-\begin_inset Formula $(a',x')\in B(a,\frac{\varepsilon}{|a|+\Vert x\Vert+1})\times B(x,\frac{\varepsilon}{|a|+\Vert x\Vert+1})$
+.
+\end_layout
+
+\begin_layout Standard
+La intersección de conjuntos convexos es convexa, y llamamos 
+\series bold
+envoltura convexa
+\series default
+ de 
+\begin_inset Formula $A\subseteq E$
 \end_inset
 
 , 
-\begin_inset Formula 
-\begin{align*}
-\Vert p(a',x')-b\Vert & =\Vert((a'-a)+a)((x'-x)+x)-ax\Vert=\\
- & =|a'-a|\Vert x'-x\Vert+|a|\Vert x'-x\Vert+|a'-a|\Vert x\Vert<\\
- & <\frac{\varepsilon}{|a|+\Vert x\Vert+1}\left(\frac{\varepsilon}{|a|+\Vert x\Vert+1}+|a|+\Vert x\Vert\right)\leq\varepsilon\frac{1+|a|+\Vert x\Vert}{|a|+\Vert x\Vert+1}=\varepsilon,
-\end{align*}
+\begin_inset Formula $\text{co}A$
+\end_inset
 
+, a la intersección de todos los convexos que contienen a 
+\begin_inset Formula $A$
 \end_inset
 
-con lo que 
-\begin_inset Formula $p(a',x')\in B(b,\varepsilon)\subseteq A$
+, que es absolutamente convexa si 
+\begin_inset Formula $A$
+\end_inset
+
+ es equilibrado.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+espacio localmente convexo
+\series default
+ es un e.v.t.
+ 
+\begin_inset Formula $(E,{\cal T})$
 \end_inset
 
+ con una base de entornos de 0 formada por conjuntos convexos, y entonces
+ 
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+ es 
+\series bold
+localmente convexa
+\series default
 .
+ Todo e.l.c.
+ tiene una base de entornos del origen formada por conjuntos absolutamente
+ convexos y cerrados.
+ Un 
+\series bold
+espacio de Fréchet
+\series default
+ es un e.l.c.
+ metrizable y completo.
 \end_layout
 
+\begin_layout Standard
+Dados un espacio vectorial 
+\begin_inset Formula $E$
 \end_inset
 
+ y 
+\begin_inset Formula $A\subseteq E$
+\end_inset
 
+ absorbente, llamamos 
+\series bold
+funcional de Minkowski
+\series default
+ asociado a 
+\begin_inset Formula $A$
+\end_inset
+
+ a 
+\begin_inset Formula $p_{A}:E\to\mathbb{R}$
+\end_inset
+
+ como 
+\begin_inset Formula $p_{A}(x)\coloneqq\inf\{t>0\mid x\in tA\}$
+\end_inset
+
+, y entonces:
 \end_layout
 
 \begin_layout Enumerate
-\begin_inset Formula $s_{y}:X\to X$
+\begin_inset Formula $p_{A}$
 \end_inset
 
- con 
-\begin_inset Formula $y\in X$
+ es no negativa y positivamente homogénea.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es convexo, 
+\begin_inset Formula $p_{A}$
+\end_inset
+
+ es subaditiva y 
+\begin_inset Formula $\{x\in E\mid p_{A}(x)<1\}\subseteq A\subseteq\{x\in E\mid p_{A}(x)\leq1\}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es absolutamente convexo, 
+\begin_inset Formula $p_{A}$
+\end_inset
+
+ es una seminorma.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Toda seminorma 
+\begin_inset Formula $p:E\to\mathbb{R}$
+\end_inset
+
+ es el funcional de Minkowski asociado a 
+\begin_inset Formula $\{x\in E\mid p(x)\leq1\}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
 \end_inset
 
+ es un e.v.t.
  y 
-\begin_inset Formula $p_{a}:X\to X$
+\begin_inset Formula $C\subseteq E$
 \end_inset
 
- con 
-\begin_inset Formula $a\in\mathbb{K}^{*}$
+ es convexo y absorbente, 
+\begin_inset Formula $0\in\mathring{C}$
 \end_inset
 
- dados por 
-\begin_inset Formula $s_{y}(x)\coloneqq x+y$
+ si y sólo si el funcional de Minkowski 
+\begin_inset Formula $p_{C}$
+\end_inset
+
+ es continuo en 
+\begin_inset Formula $E$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $\mathring{C}=\{x\in E\mid p_{C}(x)<1\}$
 \end_inset
 
  y 
-\begin_inset Formula $p_{a}(x)\coloneqq ax$
+\begin_inset Formula $\overline{C}=\{x\in E\mid p_{C}(x)\leq1\}$
 \end_inset
 
- son homeomorfismos.
-\begin_inset Note Comment
+.
+\begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-\begin_inset Formula $s_{y}$
+nproof
+\end_layout
+
 \end_inset
 
- es la composición de 
-\begin_inset Formula $x\mapsto(x,y)$
+
+\end_layout
+
+\begin_layout Standard
+Una seminorma 
+\begin_inset Formula $p:E\to\mathbb{R}$
 \end_inset
 
- con la suma, por lo que es continua, y análogamente lo es 
-\begin_inset Formula $p_{a}$
+ es continua si y sólo si 
+\begin_inset Formula $\{x\in E\mid p(x)<1\}$
 \end_inset
 
-, pero la inversa de 
-\begin_inset Formula $s_{y}$
+ es abierta, si y sólo si 
+\begin_inset Formula $0\in\mathring{\overbrace{\{x\in E\mid p(x)<1\}}}$
 \end_inset
 
- es 
-\begin_inset Formula $s_{-y}$
+, si y sólo si 
+\begin_inset Formula $p$
 \end_inset
 
- y la de 
-\begin_inset Formula $p_{a}$
+ es continua en 0, si y sólo si existe una seminorma continua 
+\begin_inset Formula $q:E\to\mathbb{R}$
 \end_inset
 
- es 
-\begin_inset Formula $p_{a^{-1}}$
+ con 
+\begin_inset Formula $p\leq q$
 \end_inset
 
-, que también son continuas.
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
 \end_inset
@@ -532,58 +1394,263 @@ status open
 
 \end_layout
 
-\begin_layout Enumerate
-La suma de un abierto y un subconjunto cualquiera de 
-\begin_inset Formula $X$
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, un e.v.t.
+ 
+\begin_inset Formula $(E,{\cal T})$
 \end_inset
 
- es abierta en 
-\begin_inset Formula $X$
+ es localmente convexo si y sólo si 
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+ está asociada a una familia de seminormas.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados dos e.l.c.
+ 
+\begin_inset Formula $E$
+\end_inset
+
+ y 
+\begin_inset Formula $F$
+\end_inset
+
+, 
+\begin_inset Formula $T:E\to F$
+\end_inset
+
+ lineal es continua si y sólo si lo es en 0, si y sólo si para toda seminorma
+ continua 
+\begin_inset Formula $q:F\to\mathbb{R}$
+\end_inset
+
+ existe una seminorma continua 
+\begin_inset Formula $p:E\to\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $q\circ T\leq p$
 \end_inset
 
 .
-\begin_inset Note Comment
+\begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-Sean 
-\begin_inset Formula $G\subseteq X$
+nproof
+\end_layout
+
 \end_inset
 
- abierto y 
-\begin_inset Formula $A\subseteq X$
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.v.t., 
+\begin_inset Formula $E'\neq0$
+\end_inset
+
+ si y sólo si existe 
+\begin_inset Formula $U\in{\cal E}(0_{E})$
+\end_inset
+
+ convexo distinto de 
+\begin_inset Formula $E$
 \end_inset
 
 .
- Todo 
-\begin_inset Formula $p\in G+A$
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
- es de la forma 
-\begin_inset Formula $p=g+a$
+
+\end_layout
+
+\begin_layout Section
+Ejemplos de espacios localmente convexos
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $Z$
 \end_inset
 
- con 
-\begin_inset Formula $g\in G$
+ es un conjunto y 
+\begin_inset Formula $\mathbb{K}^{Z}$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio vectorial, 
+\begin_inset Formula $\{f\mapsto|f(z)|\}_{z\in Z}$
+\end_inset
+
+ es una familia de seminormas en 
+\begin_inset Formula $\mathbb{K}^{Z}$
+\end_inset
+
+ que define la 
+\series bold
+topología de convergencia puntual
+\series default
+, 
+\begin_inset Formula ${\cal T}_{\text{p}}$
+\end_inset
+
+, sobre 
+\begin_inset Formula $\mathbb{K}^{Z}$
+\end_inset
+
+, en que una base de entornos en un 
+\begin_inset Formula $f:Z\to\mathbb{K}$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+\left\{ \{g\in\mathbb{K}^{Z}\mid\forall z\in F,|f(z)-g(z)|<\varepsilon\}\right\} _{F\subseteq Z\text{ finito},\varepsilon>0}.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio topológico, llamamos 
+\begin_inset Formula $C(X)$
+\end_inset
+
+ al subespacio de 
+\begin_inset Formula $(\mathbb{K}^{X},{\cal T}_{\text{p}})$
+\end_inset
+
+ de las funciones continuas y 
+\begin_inset Formula $C_{\text{b}}(X)$
+\end_inset
+
+ al subespacio de 
+\begin_inset Formula $(\mathbb{K}^{X},{\cal T}_{\text{p}})$
+\end_inset
+
+ de las funciones continuas y acotadas.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $X$
+\end_inset
+
+ es 
+\series bold
+completamente regular
+\series default
+ si para todo cerrado 
+\begin_inset Formula $A\subseteq X$
 \end_inset
 
  y 
-\begin_inset Formula $a\in A$
+\begin_inset Formula $x\in X\setminus A$
 \end_inset
 
-, pero entonces 
-\begin_inset Formula $G+a\subseteq G+A$
+ existe 
+\begin_inset Formula $f:X\to\mathbb{R}$
 \end_inset
 
- es un entorno de 
-\begin_inset Formula $g+a$
+ continua con 
+\begin_inset Formula $f(A)=0$
 \end_inset
 
- por el homeomorfismo 
-\begin_inset Formula $s_{a}$
+ y 
+\begin_inset Formula $f(x)=1$
 \end_inset
 
-.
+, y entonces, si 
+\begin_inset Formula ${\cal K}$
+\end_inset
+
+ es la familia de los compactos de 
+\begin_inset Formula $X$
+\end_inset
+
+, la familia de seminormas 
+\begin_inset Formula $\{f\mapsto\max_{x\in K}|f(x)|\}_{K\in{\cal K}}$
+\end_inset
+
+ en 
+\begin_inset Formula $C(X)$
+\end_inset
+
+ tiene asociada una topología 
+\begin_inset Formula ${\cal T}_{\text{K}}$
+\end_inset
+
+, la 
+\series bold
+topología de convergencia uniforme sobre compactos
+\series default
+, en que una base de entornos de 
+\begin_inset Formula $f\in C(X)$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+\left\{ \{g\in C(X)\mid\forall x\in K,|f(x)-g(x)|<\varepsilon\}\right\} _{K\in{\cal K},\varepsilon>0}.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
 \end_inset
@@ -591,74 +1658,234 @@ Sean
 
 \end_layout
 
-\begin_layout Enumerate
-La suma de un cerrado y un compacto de 
+\begin_layout Standard
+Una 
+\series bold
+sucesión exhaustiva de compactos
+\series default
+ de un espacio topológico 
 \begin_inset Formula $X$
 \end_inset
 
- es cerrada en 
+ es una sucesión 
+\begin_inset Formula $(K_{n})_{n}$
+\end_inset
+
+ de compactos con unión 
 \begin_inset Formula $X$
 \end_inset
 
+ y tal que cada 
+\begin_inset Formula $K_{n}\subseteq\mathring{K}_{n+1}$
+\end_inset
+
 .
-\begin_inset Note Comment
+ Todo abierto 
+\begin_inset Formula $\Omega\subseteq\mathbb{K}^{k}$
+\end_inset
+
+ es completamente regular y admite una sucesión exhaustiva de compactos
+ 
+\begin_inset Formula $(K_{n})_{n}$
+\end_inset
+
+, y entonces 
+\begin_inset Formula ${\cal T}_{\text{K}}$
+\end_inset
+
+ es la topología asociada a la familia 
+\begin_inset Formula $\{f\mapsto\max_{x\in K_{n}}|f(x)|\}_{n}$
+\end_inset
+
+ y está asociada a la métrica
+\begin_inset Formula 
+\[
+d(f,g)\coloneqq\sum_{n}\frac{1}{2^{n}}\frac{p_{K_{n}}(f-g)}{1+p_{K_{n}}(f-g)},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $(C(\Omega),{\cal T}_{\text{K}})$
+\end_inset
+
+ es un espacio de Fréchet.
+\begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-Sean 
-\begin_inset Formula $F$
+nproof
+\end_layout
+
 \end_inset
 
- el cerrado y 
-\begin_inset Formula $K$
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Weierstrass:
+\series default
+ Si 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
 \end_inset
 
- el compacto, tomamos una sucesión convergente arbitraria en 
-\begin_inset Formula $F+K$
+ es abierto, el límite de una sucesión de funciones holomorfas en 
+\begin_inset Formula $({\cal C}(\Omega),{\cal T}_{\text{K}})$
 \end_inset
 
-, 
-\begin_inset Formula $(x_{n}+y_{n})_{n}$
+ es holomorfa, y en particular 
+\begin_inset Formula $({\cal H}(\Omega),{\cal T}_{\text{K}})$
 \end_inset
 
- con cada 
-\begin_inset Formula $x_{n}\in F$
+ es un espacio de Fréchet.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
- y cada 
-\begin_inset Formula $y_{n}\in K$
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{FVV2}
+\end_layout
+
 \end_inset
 
-, y 
-\begin_inset Formula $z\coloneqq\lim_{n}(x_{n}+y_{n})$
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+soporte
+\series default
+ de una función 
+\begin_inset Formula $g:\Omega\rightarrow\mathbb{C}$
 \end_inset
 
-.
- Como 
-\begin_inset Formula $K$
+ a 
+\begin_inset Formula $\text{sop}(g)\coloneqq\overline{\{g\neq0\}}$
 \end_inset
 
- es compacto, existe una subsucesión 
-\begin_inset Formula $(x_{n_{k}})_{k}$
+[...].
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
 \end_inset
 
- convergente a un 
-\begin_inset Formula $x\in K$
+
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\Omega\subseteq\mathbb{R}^{n}$
 \end_inset
 
-, luego 
-\begin_inset Formula $(y_{n_{k}})_{k}$
+ abierto:
+\end_layout
+
+\begin_layout Enumerate
+El conjunto de funciones 
+\begin_inset Formula $f:\Omega\to\mathbb{R}$
 \end_inset
 
- converge a 
-\begin_inset Formula $z-x\in F$
+ 
+\begin_inset Formula $m$
 \end_inset
 
- y por tanto 
-\begin_inset Formula $z=(z-x)+x\in F+K$
+ veces diferenciables con 
+\begin_inset Formula $\dif^{(m)}f$
+\end_inset
+
+ continua, 
+\begin_inset Formula ${\cal E}^{m}(\Omega)\coloneqq{\cal C}^{m}(\Omega)$
+\end_inset
+
+, es un espacio de Fréchet con la 
+\series bold
+topología de convergencia uniforme sobre compactos de las funciones y sus
+ derivadas hasta el grado 
+\begin_inset Formula $m$
+\end_inset
+
+
+\series default
+, dada por la familia de seminormas
+\begin_inset Formula 
+\[
+\left\{ p_{K}^{m}(f)\coloneqq\sup_{\begin{subarray}{c}
+\alpha\in\mathbb{N}^{n}\\
+|\alpha|\coloneqq\alpha_{1}+\dots+\alpha_{n}\leq m
+\end{subarray}}\sup_{x\in K}|D^{\alpha}f(x)|\right\} _{K\subseteq\Omega\text{ compacto}},
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula 
+\[
+D^{\alpha}f(x)\coloneqq\frac{\partial^{|\alpha|}f}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal E}(\Omega)\coloneqq{\cal C}^{\infty}(\Omega)\coloneqq\bigcap_{m}{\cal C}^{m}(\Omega)$
+\end_inset
+
+ es un e.l.c.
+ metrizable con la 
+\series bold
+topología de convergencia uniforme sobre compactos de las funciones y todas
+ sus derivadas
+\series default
+, dada por la familia de seminormas 
+\begin_inset Formula $\{p_{K}^{m}\}_{K\subseteq\Omega\text{ compacto},m\in\mathbb{N}}$
 \end_inset
 
 .
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
 \end_inset
@@ -667,45 +1894,184 @@ Sean
 \end_layout
 
 \begin_layout Enumerate
-Si 
-\begin_inset Formula $Y\subseteq X$
+Si para 
+\begin_inset Formula $K\subseteq\Omega$
 \end_inset
 
- es un subespacio vectorial también lo es 
-\begin_inset Formula $\overline{Y}$
+ compacto, 
+\begin_inset Formula ${\cal D}_{K}(\Omega)\coloneqq\{f\in{\cal C}^{\infty}(\Omega)\mid\text{sop}f\subseteq K\}$
+\end_inset
+
+, llamamos 
+\series bold
+base de distribuciones
+\series default
+ a 
+\begin_inset Formula ${\cal D}(\Omega)\coloneqq\bigcup_{K\subseteq\Omega\text{ compacto}}{\cal D}_{K}(\Omega)\neq0$
+\end_inset
+
+ con la topología más fina que hace continuas las inclusiones 
+\begin_inset Formula ${\cal D}_{K}(\Omega)\hookrightarrow{\cal D}(\Omega)$
 \end_inset
 
 .
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Espacios normados
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+norma
+\series default
+ es una seminorma 
+\begin_inset Formula $q$
+\end_inset
+
+ con 
+\begin_inset Formula $q^{-1}(0)=0$
+\end_inset
+
+.
+ Un 
+\series bold
+espacio normado
+\series default
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $X$
+\end_inset
+
+ con una norma 
+\begin_inset Formula $\Vert\cdot\Vert:X\to\mathbb{R}$
+\end_inset
+
+.
+ Todo espacio normado 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es un e.l.c.
+ metrizable con la distancia 
+\begin_inset Formula $(x,y)\mapsto\Vert x-y\Vert$
+\end_inset
+
+.
+ 
 \begin_inset Note Comment
 status open
 
 \begin_layout Plain Layout
-Dados 
-\begin_inset Formula $a\in\mathbb{K}$
+
+\series bold
+Demostración:
+\series default
+ Claramente es una distancia y 
+\begin_inset Formula $\{B(0,\frac{1}{n})\}_{n\in\mathbb{N}}$
+\end_inset
+
+ es una base de entornos convexos del 0.
+ Sean 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $s:X\times X\to X$
+\end_inset
+
+ la suma y 
+\begin_inset Formula $p:\mathbb{K}\times X\to X$
+\end_inset
+
+ el producto, queremos ver que 
+\begin_inset Formula $s^{-1}(A)$
 \end_inset
 
  y 
-\begin_inset Formula $x,y\in\overline{Y}$
+\begin_inset Formula $p^{-1}(A)$
 \end_inset
 
-, 
-\begin_inset Formula $x$
+ son abiertos.
+ Sean 
+\begin_inset Formula $(x,y)\in s^{-1}(A)$
 \end_inset
 
- e 
-\begin_inset Formula $y$
+ y 
+\begin_inset Formula $b\coloneqq s(x,y)$
 \end_inset
 
- son límites de sucesiones respectivas 
-\begin_inset Formula $\{x_{n}\}_{n},\{y_{n}\}_{n}\subseteq Y$
+, existe 
+\begin_inset Formula $\varepsilon>0$
 \end_inset
 
-, con lo que 
-\begin_inset Formula $x+y=\lim_{n}(x_{n}+y_{n})\in\overline{Y}$
+ tal que 
+\begin_inset Formula $B(b,\varepsilon)\subseteq A$
+\end_inset
+
+, pero entonces, para 
+\begin_inset Formula $(x',y')\in B(x,\frac{\varepsilon}{2})\times B(y,\frac{\varepsilon}{2})$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+\Vert s(x',y')-b\Vert=\Vert\cancel{b}+(x'-x)+(y'-y)\cancel{-b}\Vert\leq\Vert x'-x\Vert+\Vert y'-y\Vert<\varepsilon,
+\]
+
+\end_inset
+
+luego 
+\begin_inset Formula $s(x',y')\in B(x,\frac{\varepsilon}{2})\subseteq A$
+\end_inset
+
+.
+ Sean 
+\begin_inset Formula $(a,x)\subseteq p^{-1}(A)$
 \end_inset
 
  y 
-\begin_inset Formula $ax=\lim_{n}ax_{n}\in\overline{Y}$
+\begin_inset Formula $b\coloneqq p(a,x)$
+\end_inset
+
+, existe 
+\begin_inset Formula $\varepsilon\in(0,1)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $B(b,\varepsilon)\subseteq A$
+\end_inset
+
+, pero entonces para 
+\begin_inset Formula $(a',x')\in B(a,\frac{\varepsilon}{|a|+\Vert x\Vert+1})\times B(x,\frac{\varepsilon}{|a|+\Vert x\Vert+1})$
+\end_inset
+
+, 
+\begin_inset Formula 
+\begin{align*}
+\Vert p(a',x')-b\Vert & =\Vert((a'-a)+a)((x'-x)+x)-ax\Vert=\\
+ & =|a'-a|\Vert x'-x\Vert+|a|\Vert x'-x\Vert+|a'-a|\Vert x\Vert<\\
+ & <\frac{\varepsilon}{|a|+\Vert x\Vert+1}\left(\frac{\varepsilon}{|a|+\Vert x\Vert+1}+|a|+\Vert x\Vert\right)\leq\varepsilon\frac{1+|a|+\Vert x\Vert}{|a|+\Vert x\Vert+1}=\varepsilon,
+\end{align*}
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $p(a',x')\in B(b,\varepsilon)\subseteq A$
 \end_inset
 
 .
@@ -716,70 +2082,140 @@ Dados
 
 \end_layout
 
-\begin_layout Enumerate
-Un subespacio vectorial de 
-\begin_inset Formula $X$
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, un e.l.c.
+ 
+\begin_inset Formula $(E,{\cal T})$
 \end_inset
 
- es propio si y sólo si su interior es vacío.
-\begin_inset Note Comment
-status open
+ es metrizable si y sólo si es 1AN, si y sólo si 
+\begin_inset Formula ${\cal T}$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Argument item:1
+ es asociada a una familia numerable de seminormas continuas.
+\begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-\begin_inset Formula $\implies]$
+nproof
+\end_layout
+
 \end_inset
 
 
 \end_layout
 
+\begin_layout Standard
+Un e.l.c.
+ 
+\begin_inset Formula $(E,{\cal T})$
 \end_inset
 
-Sea 
-\begin_inset Formula $Y<X$
+ es 
+\series bold
+normable
+\series default
+ si 
+\begin_inset Formula ${\cal T}$
 \end_inset
 
- un subespacio vectorial propio y 
-\begin_inset Formula $p\in X\setminus Y$
+ es la topología asociada a una norma en 
+\begin_inset Formula $E$
 \end_inset
 
-, para 
-\begin_inset Formula $y\in Y$
+.
+ Si 
+\begin_inset Formula $E$
 \end_inset
 
-, 
-\begin_inset Formula $(y+\frac{p}{n})_{n\in\mathbb{N}^{*}}$
+ es un e.l.c., 
+\begin_inset Formula $A\subseteq E$
 \end_inset
 
- es una sucesión de elementos de 
-\begin_inset Formula $X\setminus Y$
+ es 
+\series bold
+acotado
+\series default
+ si 
+\begin_inset Formula $\forall U\in{\cal E}(0),\exists\rho>0:A\subseteq\rho U$
 \end_inset
 
- que converge a 
-\begin_inset Formula $y$
+, si y sólo si para toda seminorma 
+\begin_inset Formula $p:E\to\mathbb{R}$
 \end_inset
 
-, con lo que 
-\begin_inset Formula $y\notin\text{int}Y$
+ continua es 
+\begin_inset Formula $\sup\{p(x)\}_{x\in A}<\infty$
 \end_inset
 
- e 
-\begin_inset Formula $\text{int}Y=\emptyset$
+.
+ 
+\series bold
+Teorema de Kolmogoroff:
+\series default
+ Un e.l.c.
+ es normable si y sólo si 
+\begin_inset Formula $0_{E}$
 \end_inset
 
-.
+ tiene un entorno acotado.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
-\begin_deeper
-\begin_layout Enumerate
-\begin_inset Argument item:1
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio normado, llamamos 
+\begin_inset Formula $B_{X}\coloneqq B[0,1]=\overline{B(0,1)}=\{x\in X\mid\Vert x\Vert\leq1\}$
+\end_inset
+
+, que es equilibrado y absorbente, y conjunto de 
+\series bold
+vectores unitarios
+\series default
+ a 
+\begin_inset Formula $S_{X}\coloneqq\partial B(0,1)=\{x\in X\mid\Vert x\Vert=1\}$
+\end_inset
+
+.
+ La norma es uniformemente continua
+\begin_inset Note Comment
 status open
 
 \begin_layout Plain Layout
-\begin_inset Formula $\impliedby]$
+, pues para 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, si 
+\begin_inset Formula $x,y\in X$
+\end_inset
+
+ cumplen 
+\begin_inset Formula $\Vert x-y\Vert<\varepsilon$
+\end_inset
+
+, por subaditividad es 
+\begin_inset Formula $\Vert x\Vert\leq\Vert x-y\Vert+\Vert y\Vert$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\left|\Vert x\Vert-\Vert y\Vert\right|=\Vert x\Vert-\Vert y\Vert\leq\Vert x-y\Vert<\varepsilon$
 \end_inset
 
 
@@ -787,13 +2223,29 @@ status open
 
 \end_inset
 
-El contrarrecíproco es trivial.
+.
+ Todo subespacio vectorial de un espacio normado es normado con la norma
+ inducida.
 \end_layout
 
-\end_deeper
+\begin_layout Standard
+Un 
+\series bold
+espacio de Banach
+\series default
+ es un espacio normado completo.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
 \end_inset
 
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
 
+-espacio normado:
 \end_layout
 
 \begin_layout Enumerate
@@ -993,90 +2445,25 @@ Toda sucesión de Cauchy en
 
 \end_layout
 
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{samepage}
-\end_layout
-
-\end_inset
-
-
+\begin_layout Section
+Operadores
 \end_layout
 
 \begin_layout Standard
-Dado un espacio normado 
-\begin_inset Formula $X$
-\end_inset
-
-, 
-\begin_inset Formula $A\subseteq X$
-\end_inset
-
- es 
+Un operador entre espacios normados se dice 
 \series bold
 acotado
 \series default
- si 
-\begin_inset Formula $\{\Vert x\Vert\}_{x\in A}$
-\end_inset
-
- está acotado superiormente.
-\end_layout
-
-\begin_layout Standard
-Dados dos 
-\begin_inset Formula $\mathbb{K}$
-\end_inset
-
--espacios normados 
-\begin_inset Formula $X$
-\end_inset
-
- e 
-\begin_inset Formula $Y$
-\end_inset
-
-, un 
-\series bold
-operador
-\series default
- de 
-\begin_inset Formula $X$
-\end_inset
-
- a 
-\begin_inset Formula $Y$
-\end_inset
-
- es una función lineal de 
+ si es continuo, y si 
 \begin_inset Formula $X$
 \end_inset
 
- a 
-\begin_inset Formula $Y$
-\end_inset
-
-, y se llama 
-\series bold
-acotado
-\series default
- si es continuo.
- Llamamos 
-\begin_inset Formula ${\cal L}(X,Y)$
-\end_inset
-
- al conjunto de operadores acotados de 
-\begin_inset Formula $X$
+ es un 
+\begin_inset Formula $\mathbb{K}$
 \end_inset
 
- a 
-\begin_inset Formula $Y$
+-espacio normado, llamamos 
+\begin_inset Formula $X^{*}\coloneqq X'={\cal L}(X,\mathbb{K})$
 \end_inset
 
 .
@@ -1298,6 +2685,19 @@ tomando
 \end_inset
 
  también lo es.
+ Si 
+\begin_inset Formula $Y=\mathbb{K}$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal L}(X,Y)=X^{*}$
+\end_inset
+
+ y esta norma se llama 
+\series bold
+norma dual
+\series default
+.
 \begin_inset Note Comment
 status open
 
@@ -1678,45 +3078,8 @@ Sean
 
 \end_layout
 
-\begin_layout Standard
-Una 
-\series bold
-forma lineal
-\series default
- en 
-\begin_inset Formula $X$
-\end_inset
-
- es una función lineal 
-\begin_inset Formula $X\to\mathbb{K}$
-\end_inset
-
-.
- Llamamos 
-\series bold
-dual algebraico
-\series default
- de 
-\begin_inset Formula $X$
-\end_inset
-
- al conjunto de formas lineales de 
-\begin_inset Formula $X$
-\end_inset
-
- y 
-\series bold
-dual topológico
-\series default
- de 
-\begin_inset Formula $X$
-\end_inset
-
- a 
-\begin_inset Formula $X^{*}\coloneqq{\cal L}(X,\mathbb{K})$
-\end_inset
-
-.
+\begin_layout Section
+Isomorfismos topológicos
 \end_layout
 
 \begin_layout Standard
@@ -2261,6 +3624,10 @@ luego
 
 \end_layout
 
+\begin_layout Section
+Espacios cociente
+\end_layout
+
 \begin_layout Standard
 \begin_inset ERT
 status open
@@ -3152,10 +4519,6 @@ Para la otra cota,
 
 \end_layout
 
-\begin_layout Standard
-Así:
-\end_layout
-
 \begin_layout Enumerate
 Todos los espacios normados de igual dimensión finta son topológicamente
  isomorfos.
@@ -3511,6 +4874,28 @@ end{samepage}
 \end_layout
 
 \begin_layout Standard
+Para 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $({\cal H}(\Omega),{\cal T}_{\text{K}})$
+\end_inset
+
+ es un espacio de Fréchet que no es de Banach.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Dado un espacio normado 
 \begin_inset Formula $X$
 \end_inset
@@ -3743,54 +5128,6 @@ end{reminder}
 \end_layout
 
 \begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{reminder}{FVV2}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\series bold
-soporte
-\series default
- de una función 
-\begin_inset Formula $g:\Omega\rightarrow\mathbb{C}$
-\end_inset
-
- a 
-\begin_inset Formula $\text{sop}(g)\coloneqq\overline{\{g\neq0\}}$
-\end_inset
-
-[...].
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{reminder}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
 Para 
 \begin_inset Formula $S\neq\emptyset$
 \end_inset
@@ -3815,7 +5152,11 @@ espacio de las funciones acotadas
 \begin_inset Formula $\ell^{\infty}(S)\coloneqq(\{f\in\mathbb{K}^{S}\mid\Vert f\Vert_{\infty}<\infty\},\Vert\cdot\Vert_{\infty})$
 \end_inset
 
-.
+ y 
+\series bold
+topología de convergencia uniforme
+\series default
+ a la topología asociada a esta norma.
  
 \begin_inset Note Note
 status open
@@ -3840,7 +5181,7 @@ Si además
 
 \begin_layout Enumerate
 El espacio 
-\begin_inset Formula $C_{b}(S)$
+\begin_inset Formula $C_{\text{b}}(S)$
 \end_inset
 
  de funciones 
@@ -3887,7 +5228,7 @@ se anula en el infinito
 \end_inset
 
  continuas que se anulan en el infinito es un subespacio cerrado de 
-\begin_inset Formula $C_{b}(S)$
+\begin_inset Formula $C_{\text{c}}(S)$
 \end_inset
 
 .
@@ -3909,7 +5250,7 @@ Si
 \end_inset
 
  es localmente compacto y Hausdorff, el espacio 
-\begin_inset Formula $C_{c}(S)$
+\begin_inset Formula $C_{\text{c}}(S)$
 \end_inset
 
  de funciones 
@@ -4216,7 +5557,7 @@ Si
 \end_inset
 
  con la medida de Lebesgue inducida, 
-\begin_inset Formula $C_{c}(\Omega)$
+\begin_inset Formula $C_{\text{c}}(\Omega)$
 \end_inset
 
  es denso en 
@@ -4237,6 +5578,22 @@ nproof
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{samepage}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Para 
 \begin_inset Formula $p\geq1$
 \end_inset
@@ -4318,6 +5675,22 @@ nproof
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{samepage}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 La suma de subespacios cerrados puede no ser un subespacio cerrado.
 \begin_inset Note Note
 status open
@@ -4368,7 +5741,7 @@ Para
 \end_inset
 
  compacto, 
-\begin_inset Formula ${\cal D}_{K}^{m}(\Omega)\coloneqq(\{f\in{\cal C}^{m}(\Omega)\mid\text{sop}f\subseteq K\},\Vert\cdot\Vert_{m})$
+\begin_inset Formula $({\cal D}_{K}^{m}(\Omega)\coloneqq\{f\in{\cal C}^{m}(\Omega)\mid\text{sop}f\subseteq K\},\Vert\cdot\Vert_{m})$
 \end_inset
 
  es un espacio de Banach.
diff --git a/af/n1b.lyx b/af/n1b.lyx
deleted file mode 100644
index 8338697..0000000
--- a/af/n1b.lyx
+++ /dev/null
@@ -1,6829 +0,0 @@
-#LyX 2.3 created this file. For more info see http://www.lyx.org/
-\lyxformat 544
-\begin_document
-\begin_header
-\save_transient_properties true
-\origin unavailable
-\textclass book
-\begin_preamble
-\input{../defs}
-\usepackage{commath}
-\end_preamble
-\use_default_options true
-\maintain_unincluded_children false
-\language spanish
-\language_package default
-\inputencoding auto
-\fontencoding global
-\font_roman "default" "default"
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-\output_sync 0
-\bibtex_command default
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-\spacing single
-\use_hyperref false
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-\use_package amsmath 1
-\use_package amssymb 1
-\use_package cancel 1
-\use_package esint 1
-\use_package mathdots 1
-\use_package mathtools 1
-\use_package mhchem 1
-\use_package stackrel 1
-\use_package stmaryrd 1
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-\cite_engine basic
-\cite_engine_type default
-\biblio_style plain
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\justification true
-\use_refstyle 1
-\use_minted 0
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
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-\html_math_output 0
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-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Standard
-David Hilbert (1862–1943) fue un influyente matemático alemán que formuló
- la teoría de los espacios de Hilbert.
- En 1900 publicó una lista de 23 problemas que marcarían en buena medida
- el progreso matemático en el siglo XX, y presentó 10 de ellos en el 
-\emph on
-\lang english
-International Congress of Mathematicians
-\emph default
-\lang spanish
- de París de 1900.
- Fue editor jefe de 
-\emph on
-\lang ngerman
-Mathematische Annalen
-\emph default
-\lang spanish
-, una revista matemática muy prestigiosa por casi 150 años, y tuvo discípulos
- como 
-\lang ngerman
-Alfréd Haar, Erhard Schmidt, Hugo Steihaus, Hermann Weyl o Ernst Zermelo
-\lang spanish
-.
-\end_layout
-
-\begin_layout Standard
-Dado un 
-\begin_inset Formula $\mathbb{K}$
-\end_inset
-
--espacio vectorial 
-\begin_inset Formula $H$
-\end_inset
-
-, 
-\begin_inset Formula $\langle\cdot,\cdot\rangle:H\times H\to\mathbb{K}$
-\end_inset
-
- es una 
-\series bold
-forma hermitiana
-\series default
- si para 
-\begin_inset Formula $a,b\in\mathbb{K}$
-\end_inset
-
- y 
-\begin_inset Formula $x,y,z\in H$
-\end_inset
-
- se tiene 
-\begin_inset Formula $\langle ax+by,z\rangle=a\langle x,z\rangle+b\langle y,z\rangle$
-\end_inset
-
- y 
-\begin_inset Formula $\langle x,y\rangle=\overline{\langle y,x\rangle}$
-\end_inset
-
-, y es 
-\series bold
-definida positiva
-\series default
- si para 
-\begin_inset Formula $x\in H\setminus0$
-\end_inset
-
- es 
-\begin_inset Formula $\langle x,x\rangle\in\mathbb{R}^{+}$
-\end_inset
-
-.
- Un 
-\series bold
-producto escalar
-\series default
- es una forma hermitiana definida positiva, y un 
-\series bold
-espacio prehilbertiano
-\series default
- es par formado por un espacio vectorial y un producto escalar sobre este.
-\end_layout
-
-\begin_layout Standard
-Dado un espacio prehilbertiano 
-\begin_inset Formula $(H,\langle\cdot,\cdot\rangle)$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Desigualdad de Cauchy-Schwartz:
-\series default
- 
-\begin_inset Formula $\forall x,y\in H,|\langle x,y\rangle|^{2}\leq\langle x,x\rangle\langle y,y\rangle$
-\end_inset
-
-, con igualdad si y sólo si 
-\begin_inset Formula $x$
-\end_inset
-
- e 
-\begin_inset Formula $y$
-\end_inset
-
- son linealmente dependientes.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $H$
-\end_inset
-
- es un espacio normado con la norma 
-\begin_inset Formula $\Vert x\Vert\coloneqq\sqrt{\langle x,x\rangle}$
-\end_inset
-
-, y para 
-\begin_inset Formula $x,y\in H$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert x+y\Vert=\Vert x\Vert+\Vert y\Vert\iff x=0\lor y=0\lor\exists a>0:x=ay$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $a,b\in\mathbb{K}$
-\end_inset
-
- y 
-\begin_inset Formula $x,y,z\in H$
-\end_inset
-
-, 
-\begin_inset Formula $\langle x,ay+bz\rangle=\overline{a}\langle x,y\rangle+\overline{b}\langle x,z\rangle$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $x,y\in H$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert x+y\Vert^{2}=\Vert x\Vert^{2}+\Vert y\Vert^{2}+2\text{Re}\langle x,y\rangle$
-\end_inset
-
-.
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-\begin_inset Formula $\Vert x+y\Vert^{2}=\langle x+y,x+y\rangle=\langle x,x\rangle+\langle x,y\rangle+\overline{\langle x,y\rangle}+\langle y,y\rangle$
-\end_inset
-
-.
-\end_layout
-
-\end_deeper
-\begin_layout Standard
-
-\series bold
-Identidades de polarización:
-\series default
- Si 
-\begin_inset Formula $H$
-\end_inset
-
- es un espacio prehilbertiano y 
-\begin_inset Formula $x,y\in H$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\langle x,y\rangle=\frac{1}{4}(\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2}+\text{i}\Vert x+\text{i}y\Vert^{2}-\text{i}\Vert x-\text{i}y\Vert^{2})$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $H$
-\end_inset
-
- se define sobre 
-\begin_inset Formula $\mathbb{R}$
-\end_inset
-
-, 
-\begin_inset Formula $\langle x,y\rangle=\frac{1}{4}(\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de von Neumann:
-\series default
- Un espacio normado 
-\begin_inset Formula $(X,\Vert\cdot\Vert)$
-\end_inset
-
- admite un producto escalar 
-\begin_inset Formula $\langle\cdot,\cdot\rangle$
-\end_inset
-
- en 
-\begin_inset Formula $X$
-\end_inset
-
- con 
-\begin_inset Formula $\langle x,x\rangle\equiv\Vert x\Vert^{2}$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $\Vert\cdot\Vert$
-\end_inset
-
- verifica la 
-\series bold
-ley del paralelogramo:
-\series default
-
-\begin_inset Formula 
-\[
-\forall x,y\in H,\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2}=2(\Vert x\Vert^{2}+\Vert y\Vert^{2}).
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Argument item:1
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\implies]$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-En general 
-\begin_inset Formula $\langle x,y+z\rangle=\overline{\langle y+z,x\rangle}=\overline{\langle y,x\rangle}+\overline{\langle z,x\rangle}=\langle x,y\rangle+\langle x,z\rangle$
-\end_inset
-
-, de donde
-\begin_inset Formula 
-\begin{multline*}
-\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2}=\langle x+y,x+y\rangle+\langle x-y,x-y\rangle=\\
-=\langle x,x\rangle+\langle x,y\rangle+\langle y,x\rangle+\langle y,y\rangle+\langle x,x\rangle-\langle x,y\rangle-\langle y,x\rangle+\langle y,y\rangle=2(\Vert x\Vert^{2}+\Vert y\Vert^{2}).
-\end{multline*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Argument item:1
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\impliedby]$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-Definimos 
-\begin_inset Formula $\langle\cdot,\cdot\rangle$
-\end_inset
-
- según la identidad de polarización, y queremos ver que es un producto escalar
- cuya norma es la inicial.
- Se tiene
-\begin_inset Formula 
-\begin{align*}
-\langle x,x\rangle & =\frac{1}{4}\left(\Vert2x\Vert^{2}-\Vert x-x\Vert^{2}+\text{i}\Vert x+\text{i}x\Vert^{2}-\text{i}\Vert x-\text{i}x\Vert^{2}\right)=\\
- & =\frac{1}{4}\left(4\Vert x\Vert^{2}+\text{i}|1+\text{i}|^{2}\Vert x\Vert^{2}-\text{i}|1-\text{i}|^{2}\Vert x\Vert^{2}\right)=\Vert x\Vert^{2},
-\end{align*}
-
-\end_inset
-
-y
-\begin_inset Formula 
-\begin{align*}
-4\langle x,y\rangle & =\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2}+\text{i}\Vert x+\text{i}y\Vert^{2}-\text{i}\Vert x-\text{i}y\Vert^{2}\\
- & =\Vert y+x\Vert^{2}-\Vert y-x\Vert^{2}+\text{i}\Vert y-\text{i}x\Vert-\text{i}\Vert y+\text{i}x\Vert^{2}=4\overline{\langle y,x\rangle}\\
- & =\Vert-x-y\Vert^{2}-\Vert-x+y\Vert^{2}+\text{i}\Vert-x-\text{i}y\Vert^{2}-\text{i}\Vert-x+\text{i}y\Vert^{2}=-4\langle-x,y\rangle\\
- & =\Vert\text{i}x+\text{i}y\Vert^{2}-\Vert\text{i}x-\text{i}y\Vert^{2}+\text{i}\Vert\text{i}x-y\Vert^{2}-\text{i}\Vert\text{i}x+y\Vert^{2}=4\frac{\langle\text{i}x,y\rangle}{\text{i}}.
-\end{align*}
-
-\end_inset
-
-Para ver que 
-\begin_inset Formula $\langle x+z,y\rangle=\langle x,y\rangle+\langle z,y\rangle$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{multline*}
-\Vert x+z+y\Vert^{2}-\Vert x+z-y\Vert^{2}=\left\Vert \left(x+\frac{y}{2}\right)+\left(z+\frac{y}{2}\right)\right\Vert ^{2}-\left\Vert \left(x+\frac{y}{2}\right)-\left(z+\frac{y}{2}\right)\right\Vert ^{2}=\\
-=2\left\Vert x+\frac{y}{2}\right\Vert ^{2}+2\left\Vert z+\frac{y}{2}\right\Vert ^{2}\cancel{-\Vert x-z\Vert^{2}}-2\left\Vert x-\frac{y}{2}\right\Vert ^{2}-2\left\Vert z-\frac{y}{2}\right\Vert ^{2}\cancel{+\Vert x-z\Vert^{2}},
-\end{multline*}
-
-\end_inset
-
-de donde
-\begin_inset Formula 
-\begin{eqnarray*}
-4\langle x+z,y\rangle & = & \Vert x+z+y\Vert^{2}-\Vert x+z-y\Vert^{2}+\text{i}\Vert x+z+\text{i}y\Vert^{2}-\text{i}\Vert x+z-\text{i}y\Vert^{2}\\
- & = & 2\left(\left\Vert x+\frac{y}{2}\right\Vert ^{2}+\left\Vert z+\frac{y}{2}\right\Vert ^{2}-\left\Vert x-\frac{y}{2}\right\Vert ^{2}-\left\Vert z-\frac{y}{2}\right\Vert \right)\\
- &  & +2\text{i}\left(\left\Vert x+\text{i}\frac{y}{2}\right\Vert ^{2}+\left\Vert z+\text{i}\frac{z}{2}\right\Vert ^{2}-\left\Vert x-\text{i}\frac{y}{2}\right\Vert ^{2}-\left\Vert z-\text{i}\frac{y}{2}\right\Vert ^{2}\right)\\
- & = & 8\left\langle x,\frac{y}{2}\right\rangle +8\left\langle z,\frac{y}{2}\right\rangle ,
-\end{eqnarray*}
-
-\end_inset
-
-y por tanto
-\begin_inset Formula 
-\[
-\langle x+z,y\rangle=2\left\langle x,\frac{y}{2}\right\rangle +2\left\langle z,\frac{y}{2}\right\rangle =\langle x,y\rangle+\langle z,y\rangle,
-\]
-
-\end_inset
-
-donde en la segunda igualdad hemos usado la primera igualdad con 
-\begin_inset Formula $z=0$
-\end_inset
-
- o 
-\begin_inset Formula $x=0$
-\end_inset
-
-.
- Usando esto y que 
-\begin_inset Formula $\langle-x,y\rangle$
-\end_inset
-
- es fácil ver que 
-\begin_inset Formula $\langle ax,y\rangle=a\langle x,y\rangle$
-\end_inset
-
- para 
-\begin_inset Formula $a\in\mathbb{Q}$
-\end_inset
-
-; para 
-\begin_inset Formula $a\in\mathbb{R}$
-\end_inset
-
- se usa la continuidad de la norma y por tanto del producto escalar, y para
- 
-\begin_inset Formula $a\in\mathbb{C}$
-\end_inset
-
- se usa 
-\begin_inset Formula $\langle\text{i}x,y\rangle=\text{i}\langle x,y\rangle$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula $(\ell^{\infty},\Vert\cdot\Vert_{\infty})$
-\end_inset
-
- y 
-\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{1})$
-\end_inset
-
- son espacios normados no prehilbertianos.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Dos espacios prehilbertianos 
-\begin_inset Formula $(H_{1},\langle\cdot,\cdot\rangle_{1})$
-\end_inset
-
- y 
-\begin_inset Formula $(H_{2},\langle\cdot,\cdot\rangle_{2})$
-\end_inset
-
- son 
-\series bold
-equivalentes
-\series default
- si existe un isomorfismo algebraico 
-\begin_inset Formula $T:H_{1}\to H_{2}$
-\end_inset
-
- con 
-\begin_inset Formula $\langle x,y\rangle_{1}=\langle T(x),T(y)\rangle_{2}$
-\end_inset
-
- para todo 
-\begin_inset Formula $x,y\in H_{1}$
-\end_inset
-
-, si y sólo si existe un isomorfismo isométrico entre los espacios normados.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $H$
-\end_inset
-
- es un espacio prehilbertiano, 
-\begin_inset Formula $x,y\in H$
-\end_inset
-
- son 
-\series bold
-ortogonales
-\series default
-, 
-\begin_inset Formula $x\bot y$
-\end_inset
-
-, si 
-\begin_inset Formula $\langle x,y\rangle=0$
-\end_inset
-
-.
- Decimos que 
-\begin_inset Formula $x\in H$
-\end_inset
-
- es 
-\series bold
-ortogonal
-\series default
- a 
-\begin_inset Formula $M\subseteq H$
-\end_inset
-
-, 
-\begin_inset Formula $x\bot M$
-\end_inset
-
-, si 
-\begin_inset Formula $\forall y\in M,x\bot y$
-\end_inset
-
-, y llamamos 
-\begin_inset Formula $M^{\bot}\coloneqq\{x\in H:x\bot M\}$
-\end_inset
-
-.
- Una familia 
-\begin_inset Formula $\{x_{i}\}_{i\in I}\subseteq H$
-\end_inset
-
- es 
-\series bold
-ortogonal
-\series default
- si 
-\begin_inset Formula $\forall i,j\in I,(i\neq j\implies x_{i}\bot x_{j})$
-\end_inset
-
-, y es 
-\series bold
-ortonormal
-\series default
- si además 
-\begin_inset Formula $\forall i,\Vert x_{i}\Vert=1$
-\end_inset
-
-.
- Entonces:
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Teorema de Pitágoras:
-\series default
- Si 
-\begin_inset Formula $x\bot y$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert x+y\Vert^{2}=\Vert x\Vert^{2}+\Vert y\Vert^{2}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $(x_{i})_{i\in I}$
-\end_inset
-
- es una familia ortogonal de elementos no nulos, es una familia linealmente
- independiente.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $M\subseteq H$
-\end_inset
-
-, 
-\begin_inset Formula $M^{\bot}$
-\end_inset
-
- es un subespacio cerrado de 
-\begin_inset Formula $H$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Lema de Gram-Schmidt:
-\series default
- Sean 
-\begin_inset Formula $H$
-\end_inset
-
- prehilbertiano, 
-\begin_inset Formula $\{x_{n}\}_{n}\subseteq H$
-\end_inset
-
- una familia contable linealmente independiente y 
-\begin_inset Formula $(u_{n})_{n}$
-\end_inset
-
- e 
-\begin_inset Formula $(y_{n})_{n}$
-\end_inset
-
- dadas por 
-\begin_inset Formula $u_{n}\coloneqq\frac{y_{n}}{\Vert y_{n}\Vert}$
-\end_inset
-
-, 
-\begin_inset Formula $y_{0}\coloneqq x_{0}$
-\end_inset
-
- y para 
-\begin_inset Formula $n\geq1$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-y_{n}\coloneqq x_{n}-\sum_{j<n}\langle x_{n},u_{j}\rangle u_{j},
-\]
-
-\end_inset
-
-
-\begin_inset Formula $(u_{n})_{n}$
-\end_inset
-
- es una sucesión ortonormal en 
-\begin_inset Formula $H$
-\end_inset
-
- y, para cada 
-\begin_inset Formula $n$
-\end_inset
-
-, 
-\begin_inset Formula $\text{span}\{u_{1},\dots,u_{n}\}=\text{span}\{x_{1},\dots,x_{n}\}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $M$
-\end_inset
-
- es un subespacio de dimensión finita del espacio prehilbertiano 
-\begin_inset Formula $H$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $M$
-\end_inset
-
- tiene una base algebraica formada por vectores ortonormales.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $M$
-\end_inset
-
- es equivalente a 
-\begin_inset Formula $\mathbb{K}^{\dim M}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Un 
-\series bold
-espacio de Hilbert
-\series default
- es un espacio prehilbertiano completo.
- Dado un espacio de medida 
-\begin_inset Formula $(\Omega,\Sigma,\mu)$
-\end_inset
-
-, 
-\begin_inset Formula $L^{2}(\Omega,\Sigma,\mu)$
-\end_inset
-
- es un espacio de Hilbert con
-\begin_inset Formula 
-\[
-\langle f,g\rangle\coloneqq\int_{\Omega}f\overline{g}\dif\mu,
-\]
-
-\end_inset
-
-y en particular lo son 
-\begin_inset Formula $\ell^{2}$
-\end_inset
-
- con 
-\begin_inset Formula $\langle x,y\rangle\coloneqq\sum_{n}x_{n}\overline{y_{n}}$
-\end_inset
-
- y 
-\begin_inset Formula $\ell_{n}^{2}$
-\end_inset
-
- con 
-\begin_inset Formula $\langle x,y\rangle\coloneqq\sum_{i}x_{i}\overline{y_{i}}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Son espacios prehilbertianos no completos:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $c_{00}$
-\end_inset
-
- con el producto escalar de 
-\begin_inset Formula $\ell^{2}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $C([a,b])$
-\end_inset
-
- con el producto escalar de 
-\begin_inset Formula $L^{2}([a,b])$
-\end_inset
-
- con la medida de Lebesgue, y entonces 
-\begin_inset Formula $C([a,b])$
-\end_inset
-
- es denso en 
-\begin_inset Formula $L^{2}([a,b])$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Mejor aproximación
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $X$
-\end_inset
-
- es un espacio vectorial, 
-\begin_inset Formula $A\subseteq X$
-\end_inset
-
- es 
-\series bold
-convexo
-\series default
- si 
-\begin_inset Formula $\forall\lambda\in[0,1]$
-\end_inset
-
-, 
-\begin_inset Formula $\lambda A+(1-\lambda)A\subseteq A$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $X$
-\end_inset
-
- es normado, 
-\begin_inset Formula $S\subseteq X$
-\end_inset
-
- no vacío y 
-\begin_inset Formula $x\in X$
-\end_inset
-
-, un 
-\begin_inset Formula $y\in S$
-\end_inset
-
- es un 
-\series bold
-vector de mejor aproximación
-\series default
- de 
-\begin_inset Formula $x$
-\end_inset
-
- a 
-\begin_inset Formula $S$
-\end_inset
-
- si 
-\begin_inset Formula $\Vert x-y\Vert=\min_{z\in S}\Vert x-z\Vert$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de mejor aproximación:
-\series default
- Si 
-\begin_inset Formula $H$
-\end_inset
-
- es un espacio prehilbertiano y 
-\begin_inset Formula $C\subseteq H$
-\end_inset
-
- es no vacío, convexo y completo, para cada 
-\begin_inset Formula $x\in H$
-\end_inset
-
- existe una mejor aproximación de 
-\begin_inset Formula $x$
-\end_inset
-
- a 
-\begin_inset Formula $C$
-\end_inset
-
-.
- 
-\series bold
-Demostración:
-\series default
- Podemos suponer por traslación que 
-\begin_inset Formula $x=0$
-\end_inset
-
-, y llamamos 
-\begin_inset Formula $\alpha\coloneqq\inf_{z\in C}\Vert z\Vert$
-\end_inset
-
-.
- Para la existencia tomamos una sucesión 
-\begin_inset Formula $\{y_{n}\}_{n}\subseteq C$
-\end_inset
-
- con 
-\begin_inset Formula $\lim_{n}\Vert y_{n}\Vert=\alpha$
-\end_inset
-
- y probamos que es de Cauchy, pues entonces por completitud existe 
-\begin_inset Formula $y\coloneqq\lim_{n}y_{n}\in C$
-\end_inset
-
- y por continuidad de la norma es 
-\begin_inset Formula $\Vert y\Vert=\alpha$
-\end_inset
-
-.
- Para 
-\begin_inset Formula $\varepsilon>0$
-\end_inset
-
- existe 
-\begin_inset Formula $n_{0}$
-\end_inset
-
- tal que si 
-\begin_inset Formula $n\geq n_{0}$
-\end_inset
-
- es 
-\begin_inset Formula $\Vert y_{n}\Vert^{2}<\alpha^{2}+\varepsilon$
-\end_inset
-
-, y por la ley del paralelogramo es
-\begin_inset Formula 
-\[
-\left\Vert \frac{y_{n}-y_{m}}{2}\right\Vert ^{2}=\frac{1}{2}(\Vert y_{n}\Vert^{2}+\Vert y_{m}\Vert^{2})-\left\Vert \frac{y_{n}+y_{m}}{2}\right\Vert ^{2}\leq\frac{1}{2}(\alpha^{2}+\varepsilon+\alpha^{2}+\varepsilon)-\alpha^{2}=\varepsilon,
-\]
-
-\end_inset
-
-pues por convexidad 
-\begin_inset Formula $\frac{y_{n}+y_{m}}{2}\in S$
-\end_inset
-
- y por tanto su norma es mayor o igual a 
-\begin_inset Formula $\alpha$
-\end_inset
-
-.
- Para la unicidad, si 
-\begin_inset Formula $y,z\in C$
-\end_inset
-
- cumplen 
-\begin_inset Formula $\Vert y\Vert=\Vert z\Vert=\alpha$
-\end_inset
-
-, por un argumento como el anterior,
-\begin_inset Formula 
-\[
-\left\Vert \frac{y-z}{2}\right\Vert ^{2}=\frac{1}{2}(\Vert y\Vert^{2}+\Vert z\Vert^{2})-\left\Vert \frac{y+z}{2}\right\Vert ^{2}\leq\frac{1}{2}(\alpha^{2}+\alpha^{2})-\alpha^{2}=0.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Como 
-\series bold
-teorema
-\series default
-, si 
-\begin_inset Formula $Y$
-\end_inset
-
- es un subespacio de un espacio prehilbertiano 
-\begin_inset Formula $H$
-\end_inset
-
- y 
-\begin_inset Formula $x\in H$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $y\in Y$
-\end_inset
-
- es de mejor aproximación de 
-\begin_inset Formula $x$
-\end_inset
-
- a 
-\begin_inset Formula $Y$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $x-y\bot Y$
-\end_inset
-
-.
-\end_layout
-
-\begin_deeper
-\begin_layout Enumerate
-\begin_inset Argument item:1
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\implies]$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-Para 
-\begin_inset Formula $z\in Y$
-\end_inset
-
- y 
-\begin_inset Formula $a\in\mathbb{K}$
-\end_inset
-
-, como 
-\begin_inset Formula $y-az\in Y$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\Vert x-y\Vert^{2}\leq\Vert x-y+az\Vert^{2}=\Vert x-y\Vert^{2}+2\text{Re}(a\langle z,x-y\rangle)+|a|^{2}\Vert z\Vert^{2},
-\]
-
-\end_inset
-
-luego 
-\begin_inset Formula $0\leq2\text{Re}(a\langle z,x-y\rangle)+|a|^{2}\Vert z\Vert^{2}$
-\end_inset
-
- y, haciendo 
-\begin_inset Formula $a=t\langle x-y,z\rangle$
-\end_inset
-
- con 
-\begin_inset Formula $t\in\mathbb{R}$
-\end_inset
-
-, 
-\begin_inset Formula $0\leq2t|\langle x-y,z\rangle|^{2}+t^{2}|\langle x-y,z\rangle|^{2}\Vert z\Vert^{2}$
-\end_inset
-
-.
- Si hubiera 
-\begin_inset Formula $z\in Y$
-\end_inset
-
- con 
-\begin_inset Formula $\langle x-y,z\rangle\neq0$
-\end_inset
-
-, 
-\begin_inset Formula $0\leq2t+t^{2}\Vert z\Vert^{2}$
-\end_inset
-
- para todo 
-\begin_inset Formula $t\in\mathbb{R}$
-\end_inset
-
-, pero si 
-\begin_inset Formula $\Vert z\Vert^{2}=0$
-\end_inset
-
-, esto es negativo cuando 
-\begin_inset Formula $t<0$
-\end_inset
-
-, y si 
-\begin_inset Formula $\Vert z\Vert^{2}>0$
-\end_inset
-
-, es negativo al menos cuando 
-\begin_inset Formula $t=-\frac{1}{\Vert z\Vert^{2}}\#$
-\end_inset
-
-, luego 
-\begin_inset Formula $x-y\bot z$
-\end_inset
-
- y 
-\begin_inset Formula $x-y\bot Y$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Argument item:1
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\impliedby]$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-Para 
-\begin_inset Formula $z\in Y$
-\end_inset
-
-, por el teorema de Pitágoras,
-\begin_inset Formula 
-\[
-\Vert x-z\Vert^{2}=\Vert x-y+y-z\Vert^{2}=\Vert x-y\Vert^{2}+\Vert y-z\Vert^{2}\geq\Vert x-y\Vert^{2}.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
-Si existe una mejor aproximación de 
-\begin_inset Formula $x$
-\end_inset
-
- a 
-\begin_inset Formula $Y$
-\end_inset
-
-, es única.
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Sean 
-\begin_inset Formula $y,z\in Y$
-\end_inset
-
- de mejor aproximación, como 
-\begin_inset Formula $x-y,x-z\in Y^{\bot}$
-\end_inset
-
-, su diferencia 
-\begin_inset Formula $y-z\in Y^{\bot}\cap Y$
-\end_inset
-
-, luego 
-\begin_inset Formula $\langle y-z,y-z\rangle=0$
-\end_inset
-
- e 
-\begin_inset Formula $y=z$
-\end_inset
-
-.
-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $Y$
-\end_inset
-
- es completo, hay vector de mejor aproximación.
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Por el teorema anterior (los subespacios son convexos).
-\end_layout
-
-\end_deeper
-\begin_layout Section
-Determinante de Gram
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $H$
-\end_inset
-
- prehilbertiano y 
-\begin_inset Formula $M\leq H$
-\end_inset
-
- de dimensión finita con base ortonormal 
-\begin_inset Formula $(e_{i})_{i}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $x\in H$
-\end_inset
-
- existe un único vector de aproximación de 
-\begin_inset Formula $x$
-\end_inset
-
- a 
-\begin_inset Formula $M$
-\end_inset
-
- dado por
-\begin_inset Formula 
-\[
-\sum_{i}\langle x,e_{i}\rangle e_{i}.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $d(x,M)^{2}=\Vert x\Vert^{2}-\sum_{i}|\langle x,e_{i}\rangle|^{2}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\series bold
-determinante de Gram
-\series default
- de 
-\begin_inset Formula $(x_{i})_{i=1}^{n}$
-\end_inset
-
- a
-\begin_inset Formula 
-\[
-G(x_{1},\dots,G_{n})\coloneqq\det(\langle x_{j},x_{i}\rangle)_{1\leq i\leq n}^{1\leq j\leq n}.
-\]
-
-\end_inset
-
-Como 
-\series bold
-teorema
-\series default
-, si 
-\begin_inset Formula $H$
-\end_inset
-
- es prehilbertiano, 
-\begin_inset Formula $M\leq H$
-\end_inset
-
- de dimensión finita con base 
-\begin_inset Formula $(b_{i})_{i}$
-\end_inset
-
- y 
-\begin_inset Formula $x\in H$
-\end_inset
-
-, el vector de mejor aproximación de 
-\begin_inset Formula $x$
-\end_inset
-
- a 
-\begin_inset Formula $M$
-\end_inset
-
- es
-\begin_inset Formula 
-\[
-\frac{-1}{G(b_{1},\dots,b_{n})}\begin{vmatrix}\langle x_{1},x_{1}\rangle & \langle x_{2},x_{1}\rangle & \cdots & \langle x_{n},x_{1}\rangle & \langle x,x_{1}\rangle\\
-\langle x_{1},x_{2}\rangle & \langle x_{2},x_{2}\rangle & \cdots & \langle x_{n},x_{2}\rangle & \langle x,x_{2}\rangle\\
-\vdots & \vdots & \ddots & \vdots & \vdots\\
-\langle x_{1},x_{n}\rangle & \langle x_{2},x_{n}\rangle & \cdots & \langle x_{n},x_{n}\rangle & \langle x,x_{n}\rangle\\
-x_{1} & x_{2} & \cdots & x_{n} & 0
-\end{vmatrix},
-\]
-
-\end_inset
-
-y
-\begin_inset Formula 
-\[
-d(x,M)=\sqrt{\frac{G(x_{1},\dots,x_{n},x)}{G(x_{1},\dots,x_{n})}}.
-\]
-
-\end_inset
-
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Algunas aplicaciones:
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Resolución de sistemas sobre-dimensionados por mínimos cuadrados.
-
-\series default
- Tenemos un fenómeno experimental que se puede modelar como una función
- lineal 
-\begin_inset Formula $y(x)=a_{1}x_{1}+\dots+a_{n}x_{n}$
-\end_inset
-
-, pero no conocemos los 
-\begin_inset Formula $a_{i}$
-\end_inset
-
-.
- Hacemos 
-\begin_inset Formula $m$
-\end_inset
-
- experimentos fijando un 
-\begin_inset Formula $x_{i}$
-\end_inset
-
- en cada uno y midiendo 
-\begin_inset Formula $y_{i}\coloneqq y(x_{i})$
-\end_inset
-
- para plantear un sistema de 
-\begin_inset Formula $m$
-\end_inset
-
- ecuaciones.
- Solo hacen falta 
-\begin_inset Formula $n$
-\end_inset
-
- experimentos cuidando que los 
-\begin_inset Formula $x_{i}$
-\end_inset
-
- sean linealmente independientes, pero en general conviene hacer más, 
-\begin_inset Formula $m>n$
-\end_inset
-
-.
- Como las mediciones son aproximadas, el sistema puede ser incompatible,
- por lo que se eligen los 
-\begin_inset Formula $a_{i}\in\mathbb{R}$
-\end_inset
-
- de forma que se minimice
-\begin_inset Formula 
-\[
-\sum_{i\in\mathbb{N}_{m}}\left(y_{i}-\sum_{j\in\mathbb{N}_{n}}a_{j}x_{ij}\right)^{2}=\left\Vert y-\sum_{j\in\mathbb{N}_{n}}a_{j}X_{j}\right\Vert ^{2},
-\]
-
-\end_inset
-
-donde 
-\begin_inset Formula $X_{j}\coloneqq(x_{1j},\dots,x_{mj})$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $X_{1},\dots,X_{n}$
-\end_inset
-
- son linealmente independientes, sea 
-\begin_inset Formula $M\coloneqq\text{span}\{X_{1},\dots,X_{n}\}<\mathbb{R}^{m}$
-\end_inset
-
-, buscamos el vector 
-\begin_inset Formula $Z\in M$
-\end_inset
-
- de mejor aproximación de 
-\begin_inset Formula $y$
-\end_inset
-
- en 
-\begin_inset Formula $M$
-\end_inset
-
- que, expresado respecto de la base 
-\begin_inset Formula $(X_{1},\dots,X_{n})$
-\end_inset
-
-, nos dará el vector 
-\begin_inset Formula $(a_{1},\dots,a_{n})$
-\end_inset
-
- buscado.
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Ajustes polinómicos por mínimos cuadrados.
-
-\series default
- Queremos modelar un fenómeno experimental como una función polinómica 
-\begin_inset Formula $f:[a,b]\to\mathbb{R}$
-\end_inset
-
-, y tenemos 
-\begin_inset Formula $k$
-\end_inset
-
- observaciones de la forma 
-\begin_inset Formula $f(t_{i})=y_{i}$
-\end_inset
-
- con 
-\begin_inset Formula $t_{1}<\dots<t_{k}$
-\end_inset
-
-.
- Existe un polinomio de grado máximo 
-\begin_inset Formula $k-1$
-\end_inset
-
- que cumple esto, pero muchas veces 
-\begin_inset Formula $k$
-\end_inset
-
- es muy grande y esto complica los cálculos y puede llevar al 
-\emph on
-\lang english
-overfitting
-\emph default
-\lang spanish
- o fenómeno de Runge.
- Entonces buscamos un polinomio 
-\begin_inset Formula $f$
-\end_inset
-
- de grado máximo 
-\begin_inset Formula $n$
-\end_inset
-
- bastante menor que 
-\begin_inset Formula $k-1$
-\end_inset
-
- que minimice
-\begin_inset Formula 
-\[
-\sum_{i\in\mathbb{N}_{k}}|y_{i}-f(t_{i})|^{2}=\left\Vert y-\sum_{j=0}^{n}f_{j}t^{j}\right\Vert ^{2},
-\]
-
-\end_inset
-
-donde 
-\begin_inset Formula $t^{j}\coloneqq(t_{1}^{j},\dots,t_{k}^{j})$
-\end_inset
-
-.
- Para ello, como para 
-\begin_inset Formula $k\geq2$
-\end_inset
-
- los 
-\begin_inset Formula $t^{j}$
-\end_inset
-
- son linealmente independientes, consideramos 
-\begin_inset Formula $M\coloneqq\text{span}\{1,t,t^{2},\dots,t^{n}\}<\mathbb{R}^{n+1}$
-\end_inset
-
- y buscamos la mejor aproximación de 
-\begin_inset Formula $y$
-\end_inset
-
- a 
-\begin_inset Formula $M$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Section
-Teorema de la proyección
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de la proyección:
-\series default
- Si 
-\begin_inset Formula $H$
-\end_inset
-
- es un espacio de Hilbert con un subespacio cerrado 
-\begin_inset Formula $M$
-\end_inset
-
- y 
-\begin_inset Formula $P_{M}:H\to M$
-\end_inset
-
- la 
-\series bold
-proyección ortogonal
-\series default
- de 
-\begin_inset Formula $H$
-\end_inset
-
- sobre 
-\begin_inset Formula $M$
-\end_inset
-
- que asigna a cada 
-\begin_inset Formula $x\in H$
-\end_inset
-
- la mejor aproximación de 
-\begin_inset Formula $x$
-\end_inset
-
- a 
-\begin_inset Formula $M$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $H$
-\end_inset
-
- es suma directa topológica de 
-\begin_inset Formula $M$
-\end_inset
-
- y 
-\begin_inset Formula $M^{\bot}$
-\end_inset
-
-, 
-\begin_inset Formula $P_{M}$
-\end_inset
-
- es la proyección canónica y, si 
-\begin_inset Formula $P_{M^{\bot}}:H\to M^{\bot}$
-\end_inset
-
- es la otra proyección canónica, si 
-\begin_inset Formula $M\neq0$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert P_{M}\Vert=1$
-\end_inset
-
-, y si 
-\begin_inset Formula $M^{\bot}\neq0$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert P_{M^{\bot}}\Vert=1$
-\end_inset
-
-.
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Por la definición de producto escalar, 
-\begin_inset Formula $M^{\bot}\leq H$
-\end_inset
-
-.
- Claramente 
-\begin_inset Formula $M\cap M^{\bot}=0$
-\end_inset
-
-, y para 
-\begin_inset Formula $x\in M$
-\end_inset
-
-, como 
-\begin_inset Formula $y\coloneqq P_{M}(x)$
-\end_inset
-
- cumple 
-\begin_inset Formula $x-y\bot M$
-\end_inset
-
-, 
-\begin_inset Formula $x=y+z$
-\end_inset
-
- con 
-\begin_inset Formula $y\in M$
-\end_inset
-
- y 
-\begin_inset Formula $z\coloneqq x-y\in M^{\bot}$
-\end_inset
-
-, luego 
-\begin_inset Formula $M+M^{\bot}=H$
-\end_inset
-
- y 
-\begin_inset Formula $H$
-\end_inset
-
- es suma directa algebraica de 
-\begin_inset Formula $M$
-\end_inset
-
- y 
-\begin_inset Formula $M^{\bot}$
-\end_inset
-
-.
- 
-\begin_inset Formula $P_{M}$
-\end_inset
-
- es la proyección canónica porque, si 
-\begin_inset Formula $y\in M$
-\end_inset
-
- y 
-\begin_inset Formula $z\in M^{\bot}$
-\end_inset
-
-, 
-\begin_inset Formula $(y+z)-y=z\bot M$
-\end_inset
-
-, y por unicidad de la mejor aproximación, 
-\begin_inset Formula $P_{M}(y+z)=y$
-\end_inset
-
-.
- 
-\begin_inset Formula $P_{M}$
-\end_inset
-
- y 
-\begin_inset Formula $P_{M^{\bot}}$
-\end_inset
-
- son lineales por ser proyecciones canónicas, y para 
-\begin_inset Formula $x=y+z\in S_{H}$
-\end_inset
-
- con 
-\begin_inset Formula $y\in M$
-\end_inset
-
- y 
-\begin_inset Formula $z\in M^{\bot}$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert x\Vert^{2}=\Vert y\Vert^{2}+\Vert z\Vert^{2}=\Vert P_{M}(x)\Vert^{2}+\Vert P_{M^{\bot}}(x)\Vert^{2}$
-\end_inset
-
- y 
-\begin_inset Formula $\Vert P_{M}(x)\Vert,\Vert P_{M^{\bot}}(x)\Vert\leq\Vert x\Vert=1$
-\end_inset
-
-, lo que prueba la continuidad y por tanto que 
-\begin_inset Formula $M$
-\end_inset
-
- es topológica.
- Además, si 
-\begin_inset Formula $M\neq0$
-\end_inset
-
-, existe 
-\begin_inset Formula $y\in S_{M}$
-\end_inset
-
- y 
-\begin_inset Formula $\Vert P_{M}(y)\Vert=\Vert y\Vert=1$
-\end_inset
-
-, luego 
-\begin_inset Formula $\Vert P_{M}\Vert=1$
-\end_inset
-
-, y análogamente para 
-\begin_inset Formula $M^{\bot}$
-\end_inset
-
-.
-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
-\begin_inset Formula $P_{M}(H)=M$
-\end_inset
-
-, 
-\begin_inset Formula $\ker P_{M}=M^{\bot}$
-\end_inset
-
- y 
-\begin_inset Formula $P_{M^{\bot}}=1_{H}-P_{M}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $x,y\in H$
-\end_inset
-
-, 
-\begin_inset Formula $\langle P_{M}(x),y\rangle=\langle x,P_{M}(y)\rangle$
-\end_inset
-
- y 
-\begin_inset Formula $\langle P_{M^{\bot}}(x),y\rangle=\langle x,P_{M^{\bot}}(y)\rangle$
-\end_inset
-
-.
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Si 
-\begin_inset Formula $x=x_{1}+x_{2}$
-\end_inset
-
- e 
-\begin_inset Formula $y=y_{1}+y_{2}$
-\end_inset
-
- con 
-\begin_inset Formula $x_{1},y_{1}\in M$
-\end_inset
-
- y 
-\begin_inset Formula $x_{2},y_{2}\in M^{\bot}$
-\end_inset
-
-, 
-\begin_inset Formula $\langle P_{M}(x),y\rangle=\langle x_{1},y_{1}+y_{2}\rangle=\langle x_{1},y_{1}\rangle=\langle x_{1}+x_{2},y_{1}\rangle=\langle x,P_{M}(y)\rangle$
-\end_inset
-
-, y para 
-\begin_inset Formula $P_{M^{\bot}}$
-\end_inset
-
- es análogo.
-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
-\begin_inset Formula $M^{\bot\bot}=M$
-\end_inset
-
-.
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Si 
-\begin_inset Formula $x\in M$
-\end_inset
-
-, para 
-\begin_inset Formula $y\in M^{\bot}$
-\end_inset
-
-, 
-\begin_inset Formula $\langle y,x\rangle=\overline{\langle x,y\rangle}=0$
-\end_inset
-
-, luego 
-\begin_inset Formula $x\in M^{\bot\bot}$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $x\in M^{\bot\bot}\subseteq H$
-\end_inset
-
-, sean 
-\begin_inset Formula $y\in M$
-\end_inset
-
- y 
-\begin_inset Formula $z\in M^{\bot}$
-\end_inset
-
- con 
-\begin_inset Formula $x=y+z$
-\end_inset
-
-, 
-\begin_inset Formula $0=\langle x,z\rangle=\langle y,z\rangle+\langle z,z\rangle=\langle z,z\rangle=\Vert z\Vert^{2}$
-\end_inset
-
-, luego 
-\begin_inset Formula $z=0$
-\end_inset
-
- y 
-\begin_inset Formula $x\in M$
-\end_inset
-
-.
-\end_layout
-
-\end_deeper
-\begin_layout Standard
-Esto no es cierto si 
-\begin_inset Formula $M$
-\end_inset
-
- no es cerrado ni si 
-\begin_inset Formula $H$
-\end_inset
-
- no es completo.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Un espacio normado es de Hilbert si y sólo si cada subespacio cerrado tiene
- un complementario topológico.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Un subconjunto 
-\begin_inset Formula $S$
-\end_inset
-
- de un espacio normado 
-\begin_inset Formula $(X,\Vert\cdot\Vert)$
-\end_inset
-
- es 
-\series bold
-total
-\series default
- si 
-\begin_inset Formula $\overline{\text{span}S}=X$
-\end_inset
-
-, y si 
-\begin_inset Formula $H$
-\end_inset
-
- es de Hilbert esto ocurre si y sólo si 
-\begin_inset Formula $S^{\bot}=0$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Section
-Dual de un espacio de Hilbert
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Riesz-Fréchet:
-\series default
- Dados un espacio de Hilbert 
-\begin_inset Formula $H$
-\end_inset
-
- y un operador 
-\begin_inset Formula $f:H\to\mathbb{K}$
-\end_inset
-
-, 
-\begin_inset Formula $f$
-\end_inset
-
- es acotado si y sólo si existe 
-\begin_inset Formula $y\in H$
-\end_inset
-
- con 
-\begin_inset Formula $f=\langle\cdot,y\rangle$
-\end_inset
-
-, en cuyo caso 
-\begin_inset Formula $y$
-\end_inset
-
- es único y 
-\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Argument item:1
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\implies]$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-Para la unicidad, si 
-\begin_inset Formula $f(x)=\langle x,y\rangle=\langle x,z\rangle$
-\end_inset
-
- para todo 
-\begin_inset Formula $x\in H$
-\end_inset
-
-, 
-\begin_inset Formula $\langle x,y-z\rangle=0$
-\end_inset
-
-, luego 
-\begin_inset Formula $y-z\bot H$
-\end_inset
-
- y, como 
-\begin_inset Formula $H^{\bot}=0$
-\end_inset
-
-, 
-\begin_inset Formula $y=z$
-\end_inset
-
-.
- Para la existencia, si 
-\begin_inset Formula $f=0$
-\end_inset
-
- tomamos 
-\begin_inset Formula $y=0$
-\end_inset
-
-, y en otro caso, 
-\begin_inset Formula $Y\coloneqq\ker f$
-\end_inset
-
- es un subespacio cerrado de 
-\begin_inset Formula $H$
-\end_inset
-
- y por tanto 
-\begin_inset Formula $H=Y\oplus Y^{\bot}$
-\end_inset
-
-, con 
-\begin_inset Formula $\dim Y^{\bot}=\dim\text{Im}f=1$
-\end_inset
-
-.
- Sea entonces 
-\begin_inset Formula $z\in Y^{\bot}$
-\end_inset
-
- unitario, la proyección ortogonal de un 
-\begin_inset Formula $x\in H$
-\end_inset
-
- sobre 
-\begin_inset Formula $Y^{\bot}$
-\end_inset
-
- es 
-\begin_inset Formula $\langle x,z\rangle z$
-\end_inset
-
-, luego 
-\begin_inset Formula $x-\langle x,z\rangle z\in Y$
-\end_inset
-
- y
-\begin_inset Formula 
-\[
-f(x)=f(x-\langle x,z\rangle z+\langle x,z\rangle z)=f(\langle x,z\rangle z)=\langle x,z\rangle f(z)=\langle x,\overline{f(z)}z\rangle\eqqcolon\langle x,y\rangle.
-\]
-
-\end_inset
-
-Para 
-\begin_inset Formula $x\in S_{H}$
-\end_inset
-
-, por la desigualdad de Cauchy-Schwartz, 
-\begin_inset Formula $\Vert f(x)\Vert^{2}=|\langle x,y\rangle|^{2}\leq\langle x,x\rangle\langle y,y\rangle=\Vert y\Vert^{2}$
-\end_inset
-
-, luego 
-\begin_inset Formula $\Vert f\Vert\leq\Vert y\Vert$
-\end_inset
-
-, pero 
-\begin_inset Formula $f(\frac{y}{\Vert y\Vert})=\frac{f(y)}{\Vert y\Vert}=\frac{\Vert y\Vert^{2}}{\Vert y\Vert}=\Vert y\Vert$
-\end_inset
-
-, luego 
-\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Argument item:1
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\impliedby]$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula $f\coloneqq\langle\cdot,y\rangle$
-\end_inset
-
- es lineal, y es continua por el argumento anterior que prueba que 
-\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-El teorema no es válido si 
-\begin_inset Formula $H$
-\end_inset
-
- no es completo.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $H$
-\end_inset
-
- un espacio de Hilbert y 
-\begin_inset Formula $T:H^{*}\to H$
-\end_inset
-
- que a cada 
-\begin_inset Formula $f$
-\end_inset
-
- le asocia el 
-\begin_inset Formula $y$
-\end_inset
-
- con 
-\begin_inset Formula $f=\langle\cdot,y\rangle$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $T$
-\end_inset
-
- es biyectiva, isométrica y lineal conjugada.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $H^{*}$
-\end_inset
-
- es un espacio de Hilbert con el producto escalar 
-\begin_inset Formula $\langle f,g\rangle^{*}\coloneqq\langle T(g),T(f)\rangle$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $J:H\to H^{**}$
-\end_inset
-
- dada por 
-\begin_inset Formula $J(x)(f)\coloneqq f(x)$
-\end_inset
-
- es un isomorfismo algebraico isométrico.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Dado un un 
-\begin_inset Formula $\mathbb{K}$
-\end_inset
-
--espacio vectorial 
-\begin_inset Formula $X$
-\end_inset
-
-, 
-\begin_inset Formula $B:X\times X\to\mathbb{K}$
-\end_inset
-
- es 
-\series bold
-bilineal
-\series default
- si las 
-\begin_inset Formula $B(\cdot,y)$
-\end_inset
-
- y 
-\begin_inset Formula $B(x,\cdot)$
-\end_inset
-
- son lineales, 
-\series bold
-sesquilineal
-\series default
- si las 
-\begin_inset Formula $B(\cdot,y)$
-\end_inset
-
- son lineales y las 
-\begin_inset Formula $B(x,\cdot)$
-\end_inset
-
- son lineales conjugadas, 
-\series bold
-simétrica
-\series default
- si 
-\begin_inset Formula $B(x,y)\equiv B(y,x)$
-\end_inset
-
- y 
-\series bold
-positiva
-\series default
- si 
-\begin_inset Formula $\forall x\in X,B(x,x)\geq0$
-\end_inset
-
-.
- Si además 
-\begin_inset Formula $X$
-\end_inset
-
- es normado, 
-\begin_inset Formula $B$
-\end_inset
-
- es 
-\series bold
-acotada
-\series default
- si 
-\begin_inset Formula $\exists M>0:\forall x,y\in X,|B(x,y)|\leq M\Vert x\Vert\Vert y\Vert$
-\end_inset
-
-, y es 
-\series bold
-fuertemente positiva
-\series default
- si 
-\begin_inset Formula $\exists c>0:\forall x\in X,B(x,x)\geq c\Vert x\Vert^{2}$
-\end_inset
-
-.
- 
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $B$
-\end_inset
-
- es bilineal o sesquilineal, es acotada si y sólo si es continua, y para
- todo 
-\begin_inset Formula $x$
-\end_inset
-
- e 
-\begin_inset Formula $y$
-\end_inset
-
- es 
-\begin_inset Formula $2B(x,x)+2B(y,y)=B(x+y,x+y)+B(x-y,x-y)$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Lax-Milgram:
-\series default
- Sean 
-\begin_inset Formula $H$
-\end_inset
-
- un espacio de Hilbert y 
-\begin_inset Formula $B$
-\end_inset
-
- una 
-\begin_inset Formula $H$
-\end_inset
-
--forma sesquilineal acotada y fuertemente positiva, existe un único isomorfismo
- de espacios de Hilbert 
-\begin_inset Formula $T:H\to H$
-\end_inset
-
- tal que 
-\begin_inset Formula $\forall x,y\in H,B(x,y)=\langle x,T(y)\rangle$
-\end_inset
-
-.
- 
-\series bold
-Demostración:
-\series default
- Sea
-\begin_inset Formula 
-\[
-Y\coloneqq\{y\in H\mid\exists z\in H:\langle\cdot,y\rangle=B(\cdot,z)\},
-\]
-
-\end_inset
-
-
-\begin_inset Formula $0\in Y$
-\end_inset
-
- tomando 
-\begin_inset Formula $z=0$
-\end_inset
-
- y 
-\begin_inset Formula $z$
-\end_inset
-
- está unívocamente determinado por 
-\begin_inset Formula $y$
-\end_inset
-
-, ya que si 
-\begin_inset Formula $\langle\cdot,y\rangle=B(\cdot,z)=B(\cdot,z')$
-\end_inset
-
- entonces 
-\begin_inset Formula $B(\cdot,z-z')=0$
-\end_inset
-
- y en particular 
-\begin_inset Formula $0=B(z-z',z-z')\geq c\Vert z-z'\Vert^{2}$
-\end_inset
-
- para cierto 
-\begin_inset Formula $c>0$
-\end_inset
-
- por ser 
-\begin_inset Formula $B$
-\end_inset
-
- fuertemente positiva, luego 
-\begin_inset Formula $z=z'$
-\end_inset
-
-.
- Como 
-\begin_inset Formula $\langle\cdot,\cdot\rangle$
-\end_inset
-
- y 
-\begin_inset Formula $B$
-\end_inset
-
- son sesquilineales, 
-\begin_inset Formula $Y$
-\end_inset
-
- es un espacio vectorial y 
-\begin_inset Formula $S:Y\to H$
-\end_inset
-
- que a cada 
-\begin_inset Formula $y$
-\end_inset
-
- le asocia el 
-\begin_inset Formula $z$
-\end_inset
-
- con 
-\begin_inset Formula $\langle\cdot,y\rangle=B(\cdot,z)$
-\end_inset
-
- es lineal.
- Entonces, para 
-\begin_inset Formula $y\in S_{Y}$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-c\Vert S(y)\Vert^{2}\leq B(S(y),S(y))=\langle S(y),y\rangle\in\mathbb{R}^{+},
-\]
-
-\end_inset
-
-pero por la desigualdad de Cauchy-Schwartz, 
-\begin_inset Formula $\langle S(y),y\rangle^{2}=|\langle S(y),y\rangle|^{2}\leq\Vert S(y)\Vert^{2}\Vert y\Vert^{2}$
-\end_inset
-
-, luego 
-\begin_inset Formula $c\Vert S(y)\Vert^{2}\leq\langle S(y),y\rangle\leq\Vert S(y)\Vert\Vert y\Vert=\Vert S(y)\Vert$
-\end_inset
-
- y 
-\begin_inset Formula $\Vert S(y)\Vert\leq\frac{1}{c}$
-\end_inset
-
-, con lo que 
-\begin_inset Formula $S$
-\end_inset
-
- es continua.
- Entonces, si 
-\begin_inset Formula $\{y_{n}\}_{n}\subseteq Y$
-\end_inset
-
- y existe 
-\begin_inset Formula $\lim_{n}y_{n}\eqqcolon y\in H$
-\end_inset
-
-, por continuidad de 
-\begin_inset Formula $S$
-\end_inset
-
- y de 
-\begin_inset Formula $B$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\langle x,y\rangle=\lim_{n}\langle x,y_{n}\rangle=\lim_{n}B(x,S(y_{n}))=B(x,S(y)),
-\]
-
-\end_inset
-
-luego 
-\begin_inset Formula $y\in Y$
-\end_inset
-
- e 
-\begin_inset Formula $Y$
-\end_inset
-
- es cerrado.
- Entonces, si 
-\begin_inset Formula $z\in Y^{\bot}$
-\end_inset
-
-, como 
-\begin_inset Formula $B(\cdot,z):H\to\mathbb{K}$
-\end_inset
-
- es continua, por el teorema de Riesz-Fréchet existe 
-\begin_inset Formula $w\in H$
-\end_inset
-
- con 
-\begin_inset Formula $B(\cdot,z)=\langle\cdot,w\rangle$
-\end_inset
-
-, luego 
-\begin_inset Formula $w\in Y$
-\end_inset
-
-, pero entonces 
-\begin_inset Formula $B(z,z)=\langle z,w\rangle=0$
-\end_inset
-
- y, por ser 
-\begin_inset Formula $B$
-\end_inset
-
- fuertemente positiva, 
-\begin_inset Formula $z=0$
-\end_inset
-
-, luego 
-\begin_inset Formula $Y^{\bot}=0$
-\end_inset
-
- e 
-\begin_inset Formula $Y=H$
-\end_inset
-
-.
- Para 
-\begin_inset Formula $z\in H$
-\end_inset
-
-, como 
-\begin_inset Formula $B(\cdot,z)$
-\end_inset
-
- es continua, existe 
-\begin_inset Formula $w\in H$
-\end_inset
-
- con 
-\begin_inset Formula $B(\cdot z)=\langle\cdot,w\rangle$
-\end_inset
-
- y por tanto 
-\begin_inset Formula $z=S(w)$
-\end_inset
-
-, luego 
-\begin_inset Formula $S$
-\end_inset
-
- es suprayectiva.
- Si 
-\begin_inset Formula $S(y)=0$
-\end_inset
-
-, para 
-\begin_inset Formula $x\in H$
-\end_inset
-
-, 
-\begin_inset Formula $\langle x,y\rangle=B(x,S(y))=0$
-\end_inset
-
- y por tanto 
-\begin_inset Formula $y=0$
-\end_inset
-
-, luego 
-\begin_inset Formula $S$
-\end_inset
-
- es inyectiva.
- Por tanto 
-\begin_inset Formula $S$
-\end_inset
-
- es biyectiva y 
-\begin_inset Formula $T\coloneqq S^{-1}$
-\end_inset
-
- cumple 
-\begin_inset Formula $\langle x,T(y)\rangle=B(x,y)$
-\end_inset
-
-.
- Además, para 
-\begin_inset Formula $y\in S_{H}$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert T(y)\Vert^{2}=\langle T(y),T(y)\rangle=B(T(y),y)\leq M\Vert T(y)\Vert\Vert y\Vert=M\Vert T(y)\Vert$
-\end_inset
-
-, siendo 
-\begin_inset Formula $M$
-\end_inset
-
- una cota de 
-\begin_inset Formula $B$
-\end_inset
-
-, de donde 
-\begin_inset Formula $\Vert T\Vert\leq M$
-\end_inset
-
- y, como 
-\begin_inset Formula $\Vert T^{-1}\Vert=\Vert S\Vert\leq\frac{1}{c}$
-\end_inset
-
-, 
-\begin_inset Formula $T$
-\end_inset
-
- es un isomorfismo topológico isométrico.
-\end_layout
-
-\begin_layout Standard
-En particular, dado un espacio vectorial 
-\begin_inset Formula $H$
-\end_inset
-
- con dos productos escalares 
-\begin_inset Formula $\langle\cdot,\cdot\rangle_{1}$
-\end_inset
-
- y 
-\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$
-\end_inset
-
- equivalentes que hacen a 
-\begin_inset Formula $H$
-\end_inset
-
- completo, existe un isomorfismo 
-\begin_inset Formula $T:H\to H$
-\end_inset
-
- de espacios de Hilbert con 
-\begin_inset Formula $\langle x,y\rangle_{1}=\langle x,T(y)\rangle_{2}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Dado un espacio medible 
-\begin_inset Formula $(\Omega,\Sigma)$
-\end_inset
-
- con medidas 
-\begin_inset Formula $\mu$
-\end_inset
-
- y 
-\begin_inset Formula $\nu$
-\end_inset
-
-, 
-\begin_inset Formula $\nu$
-\end_inset
-
- es 
-\series bold
-absolutamente continua
-\series default
- respecto de 
-\begin_inset Formula $\mu$
-\end_inset
-
- si 
-\begin_inset Formula $\forall A\in\Sigma,(\mu(A)=0\implies\nu(A)=0)$
-\end_inset
-
-, y es 
-\series bold
-finita
-\series default
- si 
-\begin_inset Formula $\nu(\Omega)<\infty$
-\end_inset
-
-.
- 
-\series bold
-Teorema de Radon-Nicodym:
-\series default
- Si 
-\begin_inset Formula $(\Omega,\Sigma)$
-\end_inset
-
- es un espacio medible con medidas finitas 
-\begin_inset Formula $\mu$
-\end_inset
-
- y 
-\begin_inset Formula $\nu$
-\end_inset
-
- siendo 
-\begin_inset Formula $\nu$
-\end_inset
-
- absolutamente continua respecto de 
-\begin_inset Formula $\mu$
-\end_inset
-
-, existe 
-\begin_inset Formula $g:\Omega\to[0,+\infty]$
-\end_inset
-
- 
-\begin_inset Formula $\mu$
-\end_inset
-
--integrable tal que
-\begin_inset Formula 
-\[
-\forall A\in\Sigma,\nu(A)=\int_{A}g\dif\mu.
-\]
-
-\end_inset
-
-
-\series bold
-Demostración:
-\series default
- 
-\begin_inset Formula $\sigma\coloneqq\mu+\nu$
-\end_inset
-
- es una medida finita en 
-\begin_inset Formula $X$
-\end_inset
-
- tal que 
-\begin_inset Formula $\forall A\in\Sigma,(\sigma(A)=0\iff\mu(A)=0)$
-\end_inset
-
-, y la función lineal entre espacios de Hilbert 
-\begin_inset Formula $T:L^{2}(\Omega,\Sigma,\sigma)\to\mathbb{R}$
-\end_inset
-
- dada por
-\begin_inset Formula 
-\[
-Tu\coloneqq\int_{\Omega}u\dif\mu
-\]
-
-\end_inset
-
-está bien definida y es continua porque, si 
-\begin_inset Formula $\Vert u\Vert_{L^{2}(\Omega,\Sigma,\sigma)}=1$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{align*}
-|Tu| & =\left|\int_{\Omega}u\dif\mu\right|\leq\int_{\Omega}|u|\dif\mu\leq\sqrt{\int_{\Omega}|u|^{2}\dif\mu}+\sqrt{\int_{\Omega}\dif\mu}\leq\\
- & \leq\sqrt{\int_{\Omega}|u|^{2}\dif\mu+\int_{\Omega}|u|^{2}\dif\nu}+\sqrt{\int_{\Omega}\dif\mu+\int_{\Omega}\dif\nu}=1+\sqrt{\sigma(X)}.
-\end{align*}
-
-\end_inset
-
-Por el teorema de representación de Riesz, existe 
-\begin_inset Formula $f\in L^{2}(\Omega,\Sigma,\sigma)$
-\end_inset
-
- tal que, para 
-\begin_inset Formula $u\in L^{2}(\Omega,\Sigma,\sigma)$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-Tu=\int_{\Omega}u\dif\mu=\int_{\Omega}uf\dif\sigma,
-\]
-
-\end_inset
-
-pero esta igualdad se da para cuando 
-\begin_inset Formula $u=\chi_{A}$
-\end_inset
-
- para cualquier 
-\begin_inset Formula $A\in{\cal F}$
-\end_inset
-
- y por linealidad para cualquier función 
-\begin_inset Formula $\Sigma$
-\end_inset
-
--medible simple, y por el teorema de convergencia dominada también se da
- para cualquier función 
-\begin_inset Formula $\Sigma$
-\end_inset
-
--medible no negativa en casi todo punto.
- Además, para 
-\begin_inset Formula $A\in\Sigma$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\mu(A)=\int_{\Omega}\chi_{A}f\dif\sigma=\int_{A}f\dif\sigma,
-\]
-
-\end_inset
-
-de modo que 
-\begin_inset Formula $f$
-\end_inset
-
- es 
-\begin_inset Formula $\Sigma$
-\end_inset
-
--medible y, haciendo 
-\begin_inset Formula $A=\{x\mid f(x)\leq0\}$
-\end_inset
-
- o 
-\begin_inset Formula $A=\{x\mid f(x)>1\}$
-\end_inset
-
-, vemos que 
-\begin_inset Formula $f(\omega)\in(0,1]$
-\end_inset
-
- para casi todo 
-\begin_inset Formula $\omega\in\Omega$
-\end_inset
-
-, de modo que 
-\begin_inset Formula $\frac{1}{g}$
-\end_inset
-
- es 
-\begin_inset Formula $\Sigma$
-\end_inset
-
--medible no negativa en casi todo punto y, en casi todo punto, 
-\begin_inset Formula $\frac{1}{f}f=1$
-\end_inset
-
-, con lo que para 
-\begin_inset Formula $A\in\Sigma$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\int_{A}\frac{1}{f}\dif\mu=\int_{A}\dif\sigma\implies\nu(A)=\sigma(A)-\mu(A)=\int_{A}\left(\frac{1}{f}-1\right)\dif\mu\eqqcolon\int_{A}g\dif\mu.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Problemas variacionales cuadráticos
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema principal de los problemas variacionales cuadráticos:
-\series default
- Sean 
-\begin_inset Formula $H$
-\end_inset
-
- un 
-\begin_inset Formula $\mathbb{R}$
-\end_inset
-
--espacio de Hilbert, 
-\begin_inset Formula $B$
-\end_inset
-
- una 
-\begin_inset Formula $H$
-\end_inset
-
--forma bilineal simétrica, acotada y fuertemente positiva, 
-\begin_inset Formula $b$
-\end_inset
-
- una 
-\begin_inset Formula $H$
-\end_inset
-
--forma lineal continua y 
-\begin_inset Formula $F:H\to\mathbb{R}$
-\end_inset
-
- dada por
-\begin_inset Formula 
-\[
-F(x)\coloneqq\frac{1}{2}B(x,x)-b(x),
-\]
-
-\end_inset
-
-entonces:
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $w\in H$
-\end_inset
-
-, 
-\begin_inset Formula $F$
-\end_inset
-
- alcanza su mínimo en 
-\begin_inset Formula $w$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $\forall y\in H,B(w,y)=b(y)$
-\end_inset
-
-.
-\end_layout
-
-\begin_deeper
-\begin_layout Enumerate
-\begin_inset Argument item:1
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\implies]$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-Fijado 
-\begin_inset Formula $y\in H$
-\end_inset
-
-, para 
-\begin_inset Formula $t\in\mathbb{R}$
-\end_inset
-
-
-\begin_inset Formula 
-\begin{align*}
-F(w+ty) & =\frac{1}{2}B(w+ty,w+ty)-b(w+ty)=\\
- & =\frac{1}{2}(B(w,w)+2tB(w,y)+t^{2}B(y,y))-b(w)-tb(y)=\\
- & =F(w)+t(B(w,y)-b(y))+\frac{1}{2}t^{2}B(y,y),
-\end{align*}
-
-\end_inset
-
-pero por hipótesis 
-\begin_inset Formula $F(w)\leq F(w+ty)$
-\end_inset
-
- para todo 
-\begin_inset Formula $t\in\mathbb{R}$
-\end_inset
-
-, luego 
-\begin_inset Formula $\varphi:\mathbb{R}\to\mathbb{R}$
-\end_inset
-
- dada por 
-\begin_inset Formula $\varphi(t)\coloneqq F(w+ty)$
-\end_inset
-
- tiene un mínimo en 
-\begin_inset Formula $t=0$
-\end_inset
-
- y 
-\begin_inset Formula $0=\varphi'(0)=B(w,y)-b(y)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Argument item:1
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\impliedby]$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-Para 
-\begin_inset Formula $y\in H$
-\end_inset
-
- y 
-\begin_inset Formula $t\in\mathbb{R}$
-\end_inset
-
-, 
-\begin_inset Formula 
-\[
-F(w+ty)=F(w)+\cancel{t(B(w,y)-b(y))}^{=0}+\frac{1}{2}t^{2}B(y,y)\geq F(w).
-\]
-
-\end_inset
-
-
-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
-Existe un único 
-\begin_inset Formula $w\in H$
-\end_inset
-
- en el que 
-\begin_inset Formula $F$
-\end_inset
-
- alcanza su mínimo.
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Como 
-\begin_inset Formula $B$
-\end_inset
-
- es bilineal, simétrica y fuertemente positiva, es un producto escalar sobre
- 
-\begin_inset Formula $H$
-\end_inset
-
-, y como existen 
-\begin_inset Formula $c,M>0$
-\end_inset
-
- con 
-\begin_inset Formula $c\Vert x\Vert^{2}\leq B(x,x)\leq M\Vert x\Vert^{2}$
-\end_inset
-
-, el producto escalar 
-\begin_inset Formula $B$
-\end_inset
-
- es equivalente al de 
-\begin_inset Formula $H$
-\end_inset
-
-, luego 
-\begin_inset Formula $b$
-\end_inset
-
- es continua con el producto escalar 
-\begin_inset Formula $B$
-\end_inset
-
- y por el teorema de Riesz-Fréchet existe un único 
-\begin_inset Formula $w\in H$
-\end_inset
-
- con 
-\begin_inset Formula $b=B(\cdot,w)=B(w,\cdot)$
-\end_inset
-
-, que es la condición del primer apartado.
-\end_layout
-
-\end_deeper
-\begin_layout Section
-Convolución y aproximación de funciones
-\end_layout
-
-\begin_layout Standard
-Dado un abierto 
-\begin_inset Formula $\Omega\subseteq\mathbb{R}^{n}$
-\end_inset
-
-, 
-\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$
-\end_inset
-
- es 
-\series bold
-localmente integrable
-\series default
- si 
-\begin_inset Formula $|f|$
-\end_inset
-
- es integrable en todo compacto 
-\begin_inset Formula $K\subseteq\Omega$
-\end_inset
-
-.
- Dadas dos funciones localmente integrables 
-\begin_inset Formula $f,g:\mathbb{R}^{n}\to\mathbb{R}$
-\end_inset
-
-, definimos su 
-\series bold
-producto de convolución
-\series default
- como 
-\begin_inset Formula $(f*g):D\to\mathbb{R}$
-\end_inset
-
- dada por
-\begin_inset Formula 
-\[
-(f*g)(a)\coloneqq\int_{\mathbb{R}^{n}}f(x)g(a-x)\dif x,
-\]
-
-\end_inset
-
-donde 
-\begin_inset Formula $D\coloneqq\{a\in\mathbb{R}^{n}\mid x\mapsto f(x)g(a-x)\text{ integrable}\}$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $f,g\in L^{2}(\mathbb{R}^{n})$
-\end_inset
-
-, 
-\begin_inset Formula $f*g$
-\end_inset
-
- está definida en todo 
-\begin_inset Formula $\mathbb{R}^{n}$
-\end_inset
-
- y es continua y uniformemente acotada con
-\begin_inset Formula 
-\[
-\Vert f*g\Vert_{\infty}\leq\Vert f\Vert_{2}\Vert g\Vert_{2}.
-\]
-
-\end_inset
-
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-El producto de convolución es conmutativo, y si 
-\begin_inset Formula $f*g$
-\end_inset
-
- está definida en casi todo punto, 
-\begin_inset Formula $\text{sop}(f*g)\subseteq\overline{\text{sop}(f)+\text{sop}(g)}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Una 
-\series bold
-sucesión de Dirac
-\series default
- es una sucesión 
-\begin_inset Formula $(K_{m}:\mathbb{R}^{n}\to\mathbb{R}^{\geq0})_{m}$
-\end_inset
-
- de funciones continuas con
-\begin_inset Formula 
-\[
-\int_{\mathbb{R}^{n}}K_{n}=1
-\]
-
-\end_inset
-
-y tal que
-\begin_inset Formula 
-\[
-\forall\varepsilon,\delta>0,\exists n_{0}:\forall n\geq n_{0},\int_{\mathbb{R}^{n}\setminus B(0,\delta)}K_{n}(x)\dif x<\varepsilon.
-\]
-
-\end_inset
-
-Por ejemplo, si 
-\begin_inset Formula $K:\mathbb{R}^{n}\to\mathbb{R}$
-\end_inset
-
- es continua, no negativa, con soporte compacto e integral 1, entonces 
-\begin_inset Formula $(x\mapsto m^{n}K(mx))_{m\geq1}$
-\end_inset
-
- es una sucesión de Dirac.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Las sucesiones de Dirac aproximan la 
-\series bold
-delta de Dirac
-\series default
-, una 
-\begin_inset Quotes cld
-\end_inset
-
-función extendida
-\begin_inset Quotes crd
-\end_inset
-
- con integral 1 que vale 0 en todo punto salvo en el origen en que el valor
- es infinito.
-\end_layout
-
-\begin_layout Standard
-Como 
-\series bold
-teorema
-\series default
-, si 
-\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$
-\end_inset
-
- es continua y acotada, la sucesión 
-\begin_inset Formula $(f*K_{m})_{m}$
-\end_inset
-
- tiende uniformemente a 
-\begin_inset Formula $f$
-\end_inset
-
- sobre subconjuntos compactos de 
-\begin_inset Formula $\mathbb{R}^{n}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$
-\end_inset
-
- es localmente integrable y 
-\begin_inset Formula $g\in{\cal D}^{k}(\mathbb{R}^{n})$
-\end_inset
-
-, 
-\begin_inset Formula $f*g\in{\cal C}^{k}(\mathbb{R}^{n})$
-\end_inset
-
- y para 
-\begin_inset Formula $\alpha\in\mathbb{N}^{n}$
-\end_inset
-
- con 
-\begin_inset Formula $\sum_{i}\alpha_{i}\leq k$
-\end_inset
-
- es
-\begin_inset Formula 
-\[
-\frac{\partial^{|\alpha|}(f*g)}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}=f*\left(\frac{\partial^{|\alpha|}g}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}\right),
-\]
-
-\end_inset
-
-con lo que 
-\begin_inset Formula $f*g$
-\end_inset
-
- es una regularización de 
-\begin_inset Formula $f$
-\end_inset
-
- a través de una función suave 
-\begin_inset Formula $g$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Como 
-\series bold
-teorema
-\series default
-, dado un abierto 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
-, 
-\begin_inset Formula ${\cal D}(G)$
-\end_inset
-
- es denso en 
-\begin_inset Formula $(C_{c}(G),\Vert\cdot\Vert_{\infty})$
-\end_inset
-
- y en 
-\begin_inset Formula $L^{p}(G)$
-\end_inset
-
- para todo 
-\begin_inset Formula $p\in[1,\infty)$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Para 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- abierto y 
-\begin_inset Formula $f\in L^{2}(G)$
-\end_inset
-
-, si para todo 
-\begin_inset Formula $\psi\in{\cal D}(G)$
-\end_inset
-
- es
-\begin_inset Formula 
-\[
-\int_{G}f\psi=0
-\]
-
-\end_inset
-
-entonces 
-\begin_inset Formula $f=0$
-\end_inset
-
- en casi todo punto, y en particular, si 
-\begin_inset Formula $f$
-\end_inset
-
- es continua, 
-\begin_inset Formula $f=0$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Principio de Dirichlet
-\end_layout
-
-\begin_layout Standard
-Dado un abierto 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
-, 
-\begin_inset Formula $u\in{\cal D}^{2}(G)$
-\end_inset
-
- es 
-\series bold
-armónica
-\series default
- en 
-\begin_inset Formula $G$
-\end_inset
-
- si 
-\begin_inset Formula $\triangle u\coloneqq\nabla^{2}u=0$
-\end_inset
-
- en todo punto de 
-\begin_inset Formula $G$
-\end_inset
-
-.
- Dada 
-\begin_inset Formula $g\in{\cal C}(S_{\mathbb{C}})$
-\end_inset
-
-, el 
-\series bold
-problema de Dirichlet
-\series default
- consiste en encontrar 
-\begin_inset Formula $u\in{\cal D}^{2}(\overline{B_{X}})$
-\end_inset
-
- armónica con 
-\begin_inset Formula $u|_{S_{\mathbb{C}}}=g$
-\end_inset
-
-.
- Para un abierto 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
-, llamamos 
-\begin_inset Formula ${\cal C}^{m}(\overline{G})$
-\end_inset
-
- al conjunto de funciones 
-\begin_inset Formula $u:\overline{G}\to\mathbb{R}$
-\end_inset
-
- con 
-\begin_inset Formula $u|_{G}\in{\cal C}^{m}(G)$
-\end_inset
-
- para las que las derivadas parciales de orden 
-\begin_inset Formula $m$
-\end_inset
-
- de 
-\begin_inset Formula $u$
-\end_inset
-
- en 
-\begin_inset Formula $G$
-\end_inset
-
- admiten prolongación continua a 
-\begin_inset Formula $\overline{G}$
-\end_inset
-
-.
- Escribimos 
-\begin_inset Formula $\partial_{j}u\coloneqq\frac{\partial u}{\partial j}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{samepage}
-\end_layout
-
-\end_inset
-
-Dados un abierto 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- acotado y no vacío, 
-\begin_inset Formula $f:G\to\mathbb{R}$
-\end_inset
-
- y 
-\begin_inset Formula $g:\partial G\to\mathbb{R}$
-\end_inset
-
-, el 
-\series bold
-problema de valores frontera para la ecuación de Poisson
-\series default
- consiste en encontrar 
-\begin_inset Formula $u:\overline{G}\to\mathbb{R}$
-\end_inset
-
- tal que 
-\begin_inset Formula $-\triangle u|_{G}=f$
-\end_inset
-
- y 
-\begin_inset Formula $u|_{\partial G}=g$
-\end_inset
-
-, y el 
-\series bold
-problema generalizado de valores frontera
-\series default
- consiste en encontrar 
-\begin_inset Formula $u:\overline{G}\to\mathbb{R}$
-\end_inset
-
- con 
-\begin_inset Formula $u|_{\partial G}=g$
-\end_inset
-
- y
-\begin_inset Formula 
-\[
-\forall v\in{\cal D}(G),\int_{G}\sum_{j=1}^{n}\frac{\partial u}{\partial x_{j}}\frac{\partial v}{\partial x_{j}}\dif x\int_{G}fv.
-\]
-
-\end_inset
-
-
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{samepage}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- es un abierto acotado no vacío, 
-\begin_inset Formula $f\in{\cal C}(\overline{G})$
-\end_inset
-
- y 
-\begin_inset Formula $g\in{\cal C}(\partial G)$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-Una 
-\begin_inset Formula $w\in{\cal C}^{2}(\overline{G})$
-\end_inset
-
- es solución del problema de valores frontera para la ecuación de Poisson
- y sólo si lo es del problema generalizado de valores frontera.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $w\in{\cal C}^{2}(\overline{G})$
-\end_inset
-
- es solución del problema variacional consistente en encontrar el mínimo
- de 
-\begin_inset Formula $F:\{u\in{\cal C}^{2}(\overline{G})\mid u|_{\partial G}=g\}\to\mathbb{R}$
-\end_inset
-
- dada por
-\begin_inset Formula 
-\[
-F(u)\coloneqq\frac{1}{2}\int_{G}\sum_{j=1}^{n}(\partial_{j}u(x))^{2}\dif x-\int_{G}fu,
-\]
-
-\end_inset
-
-entonces es solución de los dos problemas anteriores.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-El 
-\series bold
-teorema de integración por partes en varias variables
-\series default
- afirma que, si 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- es un abierto, 
-\begin_inset Formula $u\in{\cal C}^{1}(G)$
-\end_inset
-
- y 
-\begin_inset Formula $v\in{\cal D}(G)$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\int_{G}u\partial_{j}v=-\int_{G}(\partial_{j}u)v.
-\]
-
-\end_inset
-
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $G$
-\end_inset
-
- es un abierto de 
-\begin_inset Formula $\mathbb{R}^{n}$
-\end_inset
-
- y 
-\begin_inset Formula $u,w\in L^{2}(G)$
-\end_inset
-
-, 
-\begin_inset Formula $w$
-\end_inset
-
- es la 
-\series bold
-derivada generalizada 
-\begin_inset Formula $j$
-\end_inset
-
--ésima
-\series default
- de 
-\begin_inset Formula $u$
-\end_inset
-
-, 
-\begin_inset Formula $w=\partial_{j}u$
-\end_inset
-
-, si
-\begin_inset Formula 
-\[
-\forall v\in{\cal D}(G),\int_{G}u\partial_{j}v=-\int_{G}wv,
-\]
-
-\end_inset
-
-y para 
-\begin_inset Formula $\alpha\in\mathbb{N}^{n}$
-\end_inset
-
- llamamos 
-\begin_inset Formula $D^{\alpha}u\coloneqq\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}}u$
-\end_inset
-
-.
- 
-\end_layout
-
-\begin_layout Standard
-Para 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- abierto, 
-\begin_inset Formula $k\in\mathbb{N}$
-\end_inset
-
- y 
-\begin_inset Formula $p\in[1,\infty)$
-\end_inset
-
-, llamamos 
-\series bold
-espacio de Sobolev
-\series default
- a
-\begin_inset Formula 
-\[
-W^{k,p}(G)\coloneqq\{u\in L^{p}(G)\mid\forall\alpha\in\mathbb{N}^{n},(|\alpha|\leq k\implies\exists D^{\alpha}f\in L^{p}(G))\}.
-\]
-
-\end_inset
-
-Escribimos 
-\begin_inset Formula $W^{k}(G)\coloneqq W^{k,2}(G)$
-\end_inset
-
-, y generalmente consideramos el espacio de Sobolev 
-\begin_inset Formula $W^{1}(G)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- es abierto, definimos la relación de equivalencia en 
-\begin_inset Formula $G\to\mathbb{R}$
-\end_inset
-
- como 
-\begin_inset Formula $f\sim g\iff\{x\in G\mid f(x)\neq g(x)\}\text{ es de medida nula}$
-\end_inset
-
-, y 
-\begin_inset Formula $\langle\cdot,\cdot\rangle_{1,2}:W^{1}(G)/\sim\to\mathbb{R}$
-\end_inset
-
- dada por
-\begin_inset Formula 
-\[
-\langle\overline{u},\overline{v}\rangle_{1,2}\coloneqq\int_{G}\left(uv+\sum_{j}(\partial_{j}u)(\partial_{j}v)\right)
-\]
-
-\end_inset
-
-es un producto escalar en 
-\begin_inset Formula $W^{1}(G)/\sim$
-\end_inset
-
- que lo convierte en un espacio de Hilbert.
- Identificamos 
-\begin_inset Formula $W^{1}(G)$
-\end_inset
-
- con 
-\begin_inset Formula $W^{1}(G)/\sim$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\begin_inset Formula $H_{0}^{1}(G)$
-\end_inset
-
- al espacio de Hilbert obtenido como la clausura de 
-\begin_inset Formula ${\cal D}(G)$
-\end_inset
-
- en 
-\begin_inset Formula $W^{1}(G)$
-\end_inset
-
-, que en general es un subespacio propio de 
-\begin_inset Formula $W^{1}(G)$
-\end_inset
-
- pero es igual a 
-\begin_inset Formula $W^{1}(G)$
-\end_inset
-
- si 
-\begin_inset Formula $G=\mathbb{R}^{n}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- es un abierto acotado no vacío y 
-\begin_inset Formula $u\in W^{1}(G)$
-\end_inset
-
-, 
-\series bold
-
-\begin_inset Formula $u$
-\end_inset
-
- se anula en la frontera de 
-\begin_inset Formula $G$
-\end_inset
-
- en sentido generalizado
-\series default
-, 
-\begin_inset Formula $u=0$
-\end_inset
-
- en 
-\begin_inset Formula $\partial G$
-\end_inset
-
-, si 
-\begin_inset Formula $u\in H_{0}^{1}(G)$
-\end_inset
-
-, y para 
-\begin_inset Formula $f,g\in W^{1}(G)$
-\end_inset
-
-, 
-\series bold
-
-\begin_inset Formula $f=g$
-\end_inset
-
- en 
-\begin_inset Formula $\partial G$
-\end_inset
-
- en sentido generalizado
-\series default
- si 
-\begin_inset Formula $f-g\in H_{0}^{1}(G)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Desigualdad de Poincaré-Friedrichs:
-\series default
- Si 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- es un abierto acotado no vacío, existe 
-\begin_inset Formula $C>0$
-\end_inset
-
- tal que para 
-\begin_inset Formula $u\in H_{0}^{1}(G)$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-C\int_{G}u^{2}\leq\int_{G}\sum_{j=1}^{n}(\partial_{j}u)^{2}.
-\]
-
-\end_inset
-
-
-\series bold
-Demostración:
-\series default
- Sean 
-\begin_inset Formula $R\coloneqq\prod_{i}[a_{i},b_{i}]$
-\end_inset
-
- con 
-\begin_inset Formula $G\subseteq R$
-\end_inset
-
- y 
-\begin_inset Formula $u\in{\cal D}(G)$
-\end_inset
-
-, y vemos 
-\begin_inset Formula $u$
-\end_inset
-
- como una función en 
-\begin_inset Formula $R$
-\end_inset
-
- que se anula fuera de 
-\begin_inset Formula $G$
-\end_inset
-
- y con valor indefinido en 
-\begin_inset Formula $\partial G$
-\end_inset
-
-, para 
-\begin_inset Formula $x\in R$
-\end_inset
-
-, por la desigualdad de Cauchy-Schwartz,
-\begin_inset Formula 
-\begin{align*}
-(u(x))^{2} & =\left(\int_{a_{n}}^{x_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)\dif t\right)^{2}\leq\left(\int_{a_{n}}^{x_{n}}\dif t\right)\left(\int_{a_{n}}^{x_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\right)\leq\\
- & \leq(b_{n}-a_{n})\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t,
-\end{align*}
-
-\end_inset
-
-luego
-\begin_inset Formula 
-\begin{align*}
-\int_{G}u^{2} & =\int_{R}u^{2}\leq\int_{a_{1}}^{b_{1}}\cdots\int_{a_{n}}^{b_{n}}(b_{n}-a_{n})\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\dif x_{n}\cdots\dif x_{1}=\\
- & =(b_{n}-a_{n})^{2}\int_{a_{1}}^{b_{1}}\cdots\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\dif x_{n-1}\cdots\dif x_{1}=\\
- & =(b_{n}-a_{n})^{2}\int_{R}(\partial_{n}u)^{2}\dif x\leq(b_{n}-a_{n})^{2}\int_{R}\sum_{j}(\partial_{j}u)^{2}\dif x=(b_{n}-a_{n})^{2}\int_{G}\sum_{j}(\partial_{j}u)^{2}\dif x.
-\end{align*}
-
-\end_inset
-
-Para 
-\begin_inset Formula $u\in H_{0}^{1}(G)$
-\end_inset
-
-,existe una sucesión 
-\begin_inset Formula $\{u_{m}\}_{m}\subseteq{\cal D}(G)$
-\end_inset
-
- con 
-\begin_inset Formula $\lim_{m}\Vert u-u_{m}\Vert_{1,2}=0$
-\end_inset
-
- y por tanto 
-\begin_inset Formula $\lim_{m}\Vert u-u_{m}\Vert_{2}=\lim_{m}\Vert\partial_{j}u-\partial_{j}u_{m}\Vert_{2}=0$
-\end_inset
-
-, y tomando límites y usando que la norma 
-\begin_inset Formula $\Vert\cdot\Vert_{2}\leq\Vert\cdot\Vert_{1,2}$
-\end_inset
-
- y por tanto es continua en 
-\begin_inset Formula $W^{1}(G)$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-C\int_{G}u^{2}-\int_{G}\sum_{j}(\partial_{j}u)^{2}=C\Vert u\Vert_{2}^{2}-\sum_{j}\Vert\partial_{j}u\Vert_{2}^{2}=\lim_{m}\left(C\Vert u_{m}\Vert_{2}^{2}-\sum_{j}\Vert\partial_{j}u_{m}\Vert_{2}^{2}\right)\leq0.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Principio de Dirichlet:
-\series default
- Sean 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- un abierto acotado no vacío, 
-\begin_inset Formula $f\in L^{2}(G)$
-\end_inset
-
- y 
-\begin_inset Formula $g\in W^{1}(G)$
-\end_inset
-
-, 
-\begin_inset Formula $F:\{u\in W^{1}(G)\mid u-g\in H_{0}^{1}(G)\}\to\mathbb{R}$
-\end_inset
-
- dada por 
-\begin_inset Formula 
-\[
-F(u)\coloneqq\frac{1}{2}\int_{G}\sum_{j=1}^{n}(\partial_{j}u)^{2}-\int_{G}fu
-\]
-
-\end_inset
-
-alcanza su mínimo en un único punto, que es el único 
-\begin_inset Formula $u\in\text{Dom}f$
-\end_inset
-
- tal que
-\begin_inset Formula 
-\[
-\forall v\in H_{0}^{1}(G),\int_{G}\sum_{j=1}^{n}(\partial_{j}u)(\partial_{j}v)=\int_{G}fv
-\]
-
-\end_inset
-
-y la única solución en 
-\begin_inset Formula $\text{Dom}f$
-\end_inset
-
- del problema de valores frontera para la ecuación de Poisson 
-\begin_inset Formula $-\nabla^{2}u=f$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Demostración:
-\series default
- Para 
-\begin_inset Formula $u,v\in W^{1}(G)$
-\end_inset
-
- definimos 
-\begin_inset Formula 
-\begin{align*}
-B(u,v) & \coloneqq\int_{G}\sum_{j}(\partial_{j}u)(\partial_{j}v), & b_{0}(v) & \coloneqq\int_{G}fv, & b(v) & \coloneqq b_{0}(v)-B(v,g).
-\end{align*}
-
-\end_inset
-
-
-\begin_inset Formula $B$
-\end_inset
-
- es bilineal y simétrica, y es acotada porque
-\begin_inset Formula 
-\[
-|B(u,v)|=\left|\sum_{j}\int_{G}(\partial_{j}u)(\partial_{j}v)\right|\leq\sum_{j}\left|\int_{G}(\partial_{j}u)(\partial_{j}v)\right|\leq\sum_{j}\Vert\partial_{j}u\Vert_{2}\Vert\partial_{j}v\Vert_{2}\leq n\Vert u\Vert_{1,2}\Vert v\Vert_{1,2}.
-\]
-
-\end_inset
-
-Por la desigualdad de Poincaré-Friedrichs, existe 
-\begin_inset Formula $C>0$
-\end_inset
-
- tal que, para todo 
-\begin_inset Formula $v\in H$
-\end_inset
-
-, 
-\begin_inset Formula 
-\[
-C\int_{G}v^{2}\leq\int_{G}\sum_{j}(\partial_{j}v)^{2},
-\]
-
-\end_inset
-
-luego 
-\begin_inset Formula 
-\[
-C\Vert v\Vert_{1,2}^{2}=C\left(\int_{G}v^{2}+\sum_{j}(\partial_{j}v)^{2}\right)\leq(1+C)\int_{G}\sum_{j}(\partial_{j}v)^{2}=(1+C)B(v,v)
-\]
-
-\end_inset
-
-y 
-\begin_inset Formula $B$
-\end_inset
-
- es fuertemente positiva.
- Además, 
-\begin_inset Formula $b_{0}$
-\end_inset
-
- es lineal y es acotada por la desigualdad de Cauchy-Schwartz, y como además
- 
-\begin_inset Formula $B$
-\end_inset
-
- es bilineal y acotada, 
-\begin_inset Formula $b_{0}$
-\end_inset
-
- es lineal acotada y se dan las condiciones del teorema principal de los
- problemas variacionales cuadráticos.
- Ahora bien, si 
-\begin_inset Formula $w\coloneqq u-g\in H_{0}^{1}(G)$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{multline*}
-\frac{1}{2}B(w,w)-b(w)=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}(u-g))^{2}-\int_{G}f(u-g)+\int_{G}\sum_{j}(\partial_{j}(u-g))(\partial_{j}(g))=\\
-=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}(u-g))(\partial_{j}(u+g))-\int_{G}f(u-g)=\\
-=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}u)^{2}-\int_{G}fu+\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}g)^{2}+\int_{G}fg,
-\end{multline*}
-
-\end_inset
-
-luego minimizar 
-\begin_inset Formula $F$
-\end_inset
-
- equivale a minimizar 
-\begin_inset Formula $\frac{1}{2}B(w,w)-b(w)$
-\end_inset
-
-, y además
-\begin_inset Formula 
-\begin{multline*}
-B(w,v)=b(v)\iff B(u,v)-B(g,v)=b_{0}(v)-B(v,g)\iff B(u,v)=b_{0}(v)\iff\\
-\iff\int_{G}\sum_{j}(\partial_{j}u)(\partial_{j}v)=\int_{G}fv.
-\end{multline*}
-
-\end_inset
-
-Para la última parte, si 
-\begin_inset Formula $u_{0}$
-\end_inset
-
- cumple esta última fórmula para todo 
-\begin_inset Formula $v\in H_{0}^{1}(G)$
-\end_inset
-
-, por integración por partes,
-\begin_inset Formula 
-\[
-0=\int_{G}\sum_{j}(\partial_{j}u_{0})(\partial_{j}v)-\int_{G}fv=-\int_{G}\sum_{j}(\partial_{j}\partial_{j}u_{0})v-\int_{G}fv=-\int_{G}(\nabla^{2}u_{0}+f)v,
-\]
-
-\end_inset
-
-con lo que 
-\begin_inset Formula $(\nabla^{2}u_{0}+f)\bot H_{0}^{1}(G)$
-\end_inset
-
- y, como 
-\begin_inset Formula ${\cal D}(G)\subseteq H_{0}^{1}(G)$
-\end_inset
-
- es denso en 
-\begin_inset Formula $L^{2}(G)$
-\end_inset
-
-, 
-\begin_inset Formula $\nabla^{2}u_{0}+f=0$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Section
-Soluciones débiles
-\end_layout
-
-\begin_layout Standard
-Dados 
-\begin_inset Formula $k,n\in\mathbb{N}$
-\end_inset
-
- y 
-\begin_inset Formula $a_{\alpha}\in\mathbb{K}^{n}$
-\end_inset
-
- para cada 
-\begin_inset Formula $\alpha\in\mathbb{N}^{n}$
-\end_inset
-
- con 
-\begin_inset Formula $|\alpha|<k$
-\end_inset
-
-, un 
-\series bold
-operador diferencial lineal de coeficientes constantes
-\series default
- es uno de la forma
-\begin_inset Formula 
-\[
-L\coloneqq\sum_{|\alpha|\leq k}a_{\alpha}\left(\frac{\partial}{\partial x}\right)^{\alpha}\coloneqq\sum_{|\alpha|\leq k}a_{\alpha}\frac{\partial^{|\alpha|}}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}},
-\]
-
-\end_inset
-
-y su 
-\series bold
-operador adjunto
-\series default
- es
-\begin_inset Formula 
-\[
-L^{*}\coloneqq\sum_{|\alpha|\leq k}(-1)^{|\alpha|}\overline{a_{\alpha}}\left(\frac{\partial}{\partial x}\right)^{\alpha}.
-\]
-
-\end_inset
-
-Si 
-\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
-\end_inset
-
- es abierto, 
-\begin_inset Formula $\varphi,\psi\in L^{2}(G)$
-\end_inset
-
- son de clase 
-\begin_inset Formula ${\cal C}^{k}$
-\end_inset
-
- y una de las dos tiene soporte compacto, entonces 
-\begin_inset Formula $\langle L\psi,\varphi\rangle=\langle\psi,L^{*}\varphi\rangle$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Así, si 
-\begin_inset Formula $G$
-\end_inset
-
- es un abierto en 
-\begin_inset Formula $\mathbb{R}^{n}$
-\end_inset
-
-, 
-\begin_inset Formula $f,u\in L^{2}(G)$
-\end_inset
-
- son de clase 
-\begin_inset Formula ${\cal C}^{k}$
-\end_inset
-
- y 
-\begin_inset Formula $Lu=f$
-\end_inset
-
-, entonces 
-\begin_inset Formula $\langle f,\psi\rangle=\langle u,L^{*}\psi\rangle$
-\end_inset
-
- para todo 
-\begin_inset Formula $\psi\in{\cal D}(G)$
-\end_inset
-
-.
- Para 
-\begin_inset Formula $f\in L^{2}(G)$
-\end_inset
-
-, 
-\begin_inset Formula $u\in L^{2}(G)$
-\end_inset
-
- es 
-\series bold
-solución débil
-\series default
- de la ecuación en derivadas parciales 
-\begin_inset Formula $Lu=f$
-\end_inset
-
- si para todo 
-\begin_inset Formula $\psi\in{\cal D}(G)$
-\end_inset
-
- es 
-\begin_inset Formula $\langle f,\psi\rangle=\langle u,L^{*}\psi\rangle$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $L=\od{}{x}$
-\end_inset
-
- y 
-\begin_inset Formula $u,f\in L^{2}((0,1))$
-\end_inset
-
-, 
-\begin_inset Formula $Lu=f$
-\end_inset
-
- en sentido débil si y sólo si existe 
-\begin_inset Formula $F:(0,1)\to\mathbb{R}$
-\end_inset
-
- absolutamente continua con 
-\begin_inset Formula $F=u$
-\end_inset
-
- y 
-\begin_inset Formula $F'=f$
-\end_inset
-
- para casi todo 
-\begin_inset Formula $x\in(0,1)$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-La ecuación de ondas en una dimensión,
-\begin_inset Formula 
-\[
-\left\{ \begin{array}{rlrl}
-\frac{\partial^{2}u}{\partial x^{2}}-\frac{\partial^{2}u}{\partial t^{2}} & =0, & t & \in[0,+\infty),\\
-u(x,0) & \equiv f(x), & x & \in[0,\pi],\\
-\frac{\partial u}{\partial t}(x,0) & \equiv0,
-\end{array}\right.
-\]
-
-\end_inset
-
-siendo 
-\begin_inset Formula $f:[0,\pi]\to\mathbb{R}$
-\end_inset
-
- una función lineal a trozos, admite soluciones débiles que no son soluciones
- ordinarias.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Malgrange-Ehrenpreis:
-\series default
- Sean 
-\begin_inset Formula $G$
-\end_inset
-
- un abierto acotado de 
-\begin_inset Formula $\mathbb{R}^{n}$
-\end_inset
-
- y 
-\begin_inset Formula $L$
-\end_inset
-
- un operador en derivadas parciales lineal con coeficientes constantes,
- existe un operador lineal continuo 
-\begin_inset Formula $K:L^{2}(G)\to L^{2}(G)$
-\end_inset
-
- tal que para todo 
-\begin_inset Formula $f\in L^{2}(G)$
-\end_inset
-
-, 
-\begin_inset Formula $u\coloneqq K(f)$
-\end_inset
-
- es solución débil de 
-\begin_inset Formula $Lu=f$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Demostración:
-\series default
- Definimos 
-\begin_inset Formula $\langle\varphi,\psi\rangle_{L}\coloneqq\langle L^{*}\varphi,L^{*}\psi\rangle_{2}$
-\end_inset
-
-, y para ver que es un producto escalar sobre 
-\begin_inset Formula ${\cal D}(G)$
-\end_inset
-
- vemos que existe 
-\begin_inset Formula $C>0$
-\end_inset
-
- tal que, para 
-\begin_inset Formula $\psi\in{\cal D}(G)$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert\psi\Vert_{2}\leq C\Vert L^{*}\psi\Vert_{2}$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $L^{*}=\frac{\partial}{\partial x_{1}}$
-\end_inset
-
-, llamando 
-\begin_inset Formula $\psi(x)\coloneqq0$
-\end_inset
-
- para 
-\begin_inset Formula $x\notin G$
-\end_inset
-
-, para 
-\begin_inset Formula $x\in G$
-\end_inset
-
-, como 
-\begin_inset Formula $\text{sop}\psi\subseteq G$
-\end_inset
-
- es compacto, sea 
-\begin_inset Formula $m\coloneqq\inf_{x\in G}x_{1}$
-\end_inset
-
-,
-\begin_inset Formula 
-\begin{align*}
-\psi(x)^{2} & =\left(\int_{m}^{x_{1}}\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\dif t\right)^{2}\leq\left(\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|\cdot1\dif t\right)\leq\\
- & \leq\int_{m}^{x_{1}}\dif t\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}\dif t\leq d\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2},
-\end{align*}
-
-\end_inset
-
-donde 
-\begin_inset Formula $d$
-\end_inset
-
- es el diámetro de 
-\begin_inset Formula $G$
-\end_inset
-
-, e integrando de nuevo,
-\begin_inset Formula 
-\begin{align*}
-\Vert\psi\Vert_{2}^{2} & =\int_{G}\psi(x)^{2}\dif x\leq d\int_{m}^{x_{1}}\int_{-\infty}^{x_{2}}\cdots\int_{-\infty}^{x_{n}}\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}\dif t\dif x_{n}\cdots\dif x_{1}\leq\\
- & \leq d^{2}\int_{G}\left|\frac{\partial\psi}{\partial x_{1}}(x)\right|^{2}\dif x=d^{2}\Vert L^{*}\psi\Vert_{2}^{2}.
-\end{align*}
-
-\end_inset
-
-Si 
-\begin_inset Formula $L^{*}=\frac{\partial}{\partial x_{i}}$
-\end_inset
-
- para otro 
-\begin_inset Formula $i$
-\end_inset
-
-, es análogo, y si 
-\begin_inset Formula $L^{*}=\left(\frac{\partial}{\partial x}\right)^{|\alpha|}$
-\end_inset
-
-, por inducción, 
-\begin_inset Formula $\Vert\psi\Vert_{2}\leq d^{|\alpha|}\Vert L^{*}\psi\Vert_{2}$
-\end_inset
-
-.
- Para 
-\begin_inset Formula $L$
-\end_inset
-
- arbitrario basta hacer combinaciones lineales.
- Visto esto, sean 
-\begin_inset Formula $H_{0}\coloneqq({\cal D}(G),\langle\cdot,\cdot\rangle_{L})$
-\end_inset
-
- y 
-\begin_inset Formula $H$
-\end_inset
-
- su compleción, 
-\begin_inset Formula $L^{*}:H_{0}\to L^{2}(G)$
-\end_inset
-
- es lineal y continuo y por tanto admite una extensión lineal y continua
- 
-\begin_inset Formula $\hat{L}^{*}:H\to L^{2}(G)$
-\end_inset
-
-.
- Sea ahora 
-\begin_inset Formula $f\in L^{2}(G)$
-\end_inset
-
- y 
-\begin_inset Formula $l_{0}:H_{0}\to\mathbb{K}$
-\end_inset
-
- dada por 
-\begin_inset Formula $l_{0}(\psi)\coloneqq\langle\psi,f\rangle_{2}$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-|l_{0}(\psi)|=|\langle\psi,f\rangle_{2}|\leq\Vert\psi\Vert_{2}\Vert f\Vert_{2}\leq C\Vert f\Vert_{2}\Vert L^{*}\psi\Vert_{2},
-\]
-
-\end_inset
-
-donde 
-\begin_inset Formula $C$
-\end_inset
-
- es tal que 
-\begin_inset Formula $\Vert\psi\Vert_{2}\leq C\Vert L^{*}\psi\Vert_{2}$
-\end_inset
-
- para todo 
-\begin_inset Formula $C$
-\end_inset
-
-, de modo que 
-\begin_inset Formula $l_{0}$
-\end_inset
-
- es lineal continua por la cota 
-\begin_inset Formula $C\Vert f\Vert_{2}$
-\end_inset
-
- y se puede extender a una forma lineal y continua 
-\begin_inset Formula $l:H\to\mathbb{K}$
-\end_inset
-
- con 
-\begin_inset Formula $\Vert l\Vert\leq C\Vert f\Vert_{2}$
-\end_inset
-
-.
- Por el teorema de Riesz, existe un único 
-\begin_inset Formula $\hat{u}\in H$
-\end_inset
-
- con 
-\begin_inset Formula $l(h)\equiv\langle h,\hat{u}\rangle_{L}$
-\end_inset
-
- para 
-\begin_inset Formula $h\in H$
-\end_inset
-
- y además 
-\begin_inset Formula $\Vert\hat{u}\Vert_{H}=\Vert l\Vert_{H}$
-\end_inset
-
-, y tomando 
-\begin_inset Formula $u\coloneqq\hat{L}^{*}\hat{u}$
-\end_inset
-
-, 
-\begin_inset Formula $l(h)=\langle\hat{L}^{*}h,\hat{L}^{*}\hat{u}\rangle=\langle\hat{L}^{*}h,u\rangle_{2}$
-\end_inset
-
-, pero para 
-\begin_inset Formula $\psi\in{\cal D}(G)$
-\end_inset
-
-, 
-\begin_inset Formula $l(\psi)=\langle\psi,f\rangle_{2}$
-\end_inset
-
- y 
-\begin_inset Formula $\hat{L}^{*}(\psi)=L^{*}\psi$
-\end_inset
-
-, con lo que 
-\begin_inset Formula $\langle L^{*}\psi,u\rangle_{2}=l(\psi)=\langle\psi,f\rangle_{2}$
-\end_inset
-
-, y basta llamar 
-\begin_inset Formula $K(f)\coloneqq u$
-\end_inset
-
-.
- Para la continuidad de 
-\begin_inset Formula $K$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\Vert K(f)\Vert_{2}=\Vert u\Vert_{2}=\Vert\hat{L}^{*}\hat{u}\Vert_{2}=\Vert\hat{u}\Vert_{H}=\Vert l\Vert_{H}=\sup_{\Vert\psi\Vert_{H}=\Vert L^{*}\psi\Vert_{2}=1}|l(\psi)|\leq C\Vert f\Vert_{2}.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Método de Galerkin
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $M_{1}\subseteq M_{2}\subseteq\dots\subseteq M_{n}\subseteq\dots$
-\end_inset
-
- una sucesión de subespacios cerrados de un espacio de Hilbert 
-\begin_inset Formula $H$
-\end_inset
-
- con unión densa en 
-\begin_inset Formula $H$
-\end_inset
-
-, 
-\begin_inset Formula $a:H\times H\to\mathbb{R}$
-\end_inset
-
- bilineal, simétrica, continua y fuertemente positiva, 
-\begin_inset Formula $b:H\to\mathbb{R}$
-\end_inset
-
- lineal continua,
-\begin_inset Formula 
-\[
-J(x)\coloneqq\frac{1}{2}a(x,x)-b(x)
-\]
-
-\end_inset
-
-para 
-\begin_inset Formula $x\in H$
-\end_inset
-
-, 
-\begin_inset Formula $u\in H$
-\end_inset
-
- con 
-\begin_inset Formula $J(u)$
-\end_inset
-
- mínimo y, para 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
-, 
-\begin_inset Formula $u_{n}\in M_{n}$
-\end_inset
-
- con 
-\begin_inset Formula $J(u_{n})$
-\end_inset
-
- mínimo, de modo que 
-\begin_inset Formula $a(x,u_{n})=b(x)$
-\end_inset
-
- para todo 
-\begin_inset Formula $x\in M_{n}$
-\end_inset
-
- y 
-\begin_inset Formula $a(x,u)=b(x)$
-\end_inset
-
- para todo 
-\begin_inset Formula $x\in H$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Teorema de Galerkin-Ritz:
-\series default
- 
-\begin_inset Formula $\lim_{n}u_{n}=u$
-\end_inset
-
-.
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Para 
-\begin_inset Formula $x\in M_{n}$
-\end_inset
-
-, 
-\begin_inset Formula $a(x,u_{n})=b(x)$
-\end_inset
-
-, y para 
-\begin_inset Formula $x\in H$
-\end_inset
-
-, 
-\begin_inset Formula $a(x,u)=f(x)$
-\end_inset
-
-, luego 
-\begin_inset Formula $a(x,u-u_{n})=b(x)-b(x)=0$
-\end_inset
-
- para 
-\begin_inset Formula $x\in M_{n}$
-\end_inset
-
-.
- Pero 
-\begin_inset Formula $a$
-\end_inset
-
- es un producto escalar equivalente al de 
-\begin_inset Formula $H$
-\end_inset
-
-, luego 
-\begin_inset Formula $u-u_{n}\bot M_{n}$
-\end_inset
-
- y, si 
-\begin_inset Formula $P_{n}:H\to M_{n}$
-\end_inset
-
- es la proyección ortogonal, 
-\begin_inset Formula $P_{n}(u)=u_{n}$
-\end_inset
-
-.
- Por el teorema de la proyección, 
-\begin_inset Formula $\Vert u-u_{n}\Vert=\Vert u-P_{n}(u)\Vert=d(u,M_{n})$
-\end_inset
-
-, pero por la densidad es 
-\begin_inset Formula $d(u,\bigcup_{n}M_{n})=0$
-\end_inset
-
-, y para 
-\begin_inset Formula $\varepsilon>0$
-\end_inset
-
- existen 
-\begin_inset Formula $n_{0}\in\mathbb{N}$
-\end_inset
-
- e 
-\begin_inset Formula $y\in M_{n_{0}}$
-\end_inset
-
- con 
-\begin_inset Formula $\Vert u-y\Vert<\varepsilon$
-\end_inset
-
-, y como la sucesión es creciente, para 
-\begin_inset Formula $n\geq n_{0}$
-\end_inset
-
-, 
-\begin_inset Formula $\Vert u-u_{n}\Vert=d(u,M_{n})\leq d(u,M_{n_{0}})\leq\Vert u-y\Vert<\varepsilon$
-\end_inset
-
-, con lo que 
-\begin_inset Formula $\lim_{n}u_{n}=u$
-\end_inset
-
-.
-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
-Dados 
-\begin_inset Formula $c,d>0$
-\end_inset
-
- con 
-\begin_inset Formula $a(x,y)\leq d\Vert x\Vert\Vert y\Vert$
-\end_inset
-
- y 
-\begin_inset Formula $c\Vert x\Vert^{2}\leq a(x,x)$
-\end_inset
-
- para todo 
-\begin_inset Formula $x,y\in H$
-\end_inset
-
-, 
-\begin_inset Formula $c\Vert u\Vert\leq\Vert b\Vert$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Razón de convergencia:
-\series default
- 
-\begin_inset Formula $\Vert u-u_{n}\Vert\leq\frac{d}{c}d(u,M_{n})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Estimación del error:
-\series default
- Si 
-\begin_inset Formula $\beta\leq J(x)$
-\end_inset
-
- para todo 
-\begin_inset Formula $x\in H$
-\end_inset
-
-, para 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
- es 
-\begin_inset Formula $\frac{c}{2}\Vert u-u_{n}\Vert^{2}\leq J(u_{n})-\beta$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-El 
-\series bold
-método de Galerkin
-\series default
- para resolver un problema de esta forma consiste en tomar en el teorema
- anterior los 
-\begin_inset Formula $M_{n}$
-\end_inset
-
- de dimensión finita y resolver los sistemas de ecuaciones lineales resultantes,
- con matriz de coeficientes simétrica y definida positiva de tamaño 
-\begin_inset Formula $\dim M_{n}$
-\end_inset
-
-.
- Tomando adecuadamente las bases de los 
-\begin_inset Formula $M_{n}$
-\end_inset
-
- se puede conseguir que las matrices tengan muchas entradas nulas.
-\end_layout
-
-\begin_layout Section
-Bases hilbertianas
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $(H_{i})_{i\in I}$
-\end_inset
-
- una familia de 
-\begin_inset Formula $\mathbb{K}$
-\end_inset
-
--espacios de Hilbert, 
-\begin_inset Formula $H_{0}\coloneqq\prod_{i\in I}H_{i}$
-\end_inset
-
- y 
-\begin_inset Formula $\langle\cdot,\cdot\rangle:H_{0}\times H_{0}\to[0,+\infty]$
-\end_inset
-
- dada por
-\begin_inset Formula 
-\[
-\langle x,y\rangle\coloneqq\sum_{i\in I}\langle x_{i},y_{i}\rangle_{H_{i}},
-\]
-
-\end_inset
-
-llamamos 
-\series bold
-suma directa hilbertiana
-\series default
- o 
-\series bold
-suma 
-\begin_inset Formula $\ell^{2}$
-\end_inset
-
-
-\series default
- de 
-\begin_inset Formula $\{H_{i}\}_{i\in I}$
-\end_inset
-
- al espacio de Hilbert 
-\begin_inset Formula 
-\[
-\bigoplus_{i\in I}H_{i}\coloneqq\ell^{2}((H_{i})_{i\in I})\coloneqq(\{x\in H_{0}\mid\langle x,x\rangle<\infty\},\langle\cdot,\cdot\rangle).
-\]
-
-\end_inset
-
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Cada 
-\begin_inset Formula $H_{i}$
-\end_inset
-
- es isométricamente isomorfo al subespacio de 
-\begin_inset Formula $H$
-\end_inset
-
- de los vectores con todas las coordenadas nulas salvo la 
-\begin_inset Formula $i$
-\end_inset
-
-, los 
-\begin_inset Formula $H_{i}$
-\end_inset
-
- son mutuamente ortogonales en 
-\begin_inset Formula $H$
-\end_inset
-
-, 
-\begin_inset Formula $H$
-\end_inset
-
- es la clausura lineal cerrada de los 
-\begin_inset Formula $H_{i}$
-\end_inset
-
- y cada 
-\begin_inset Formula $x\in H$
-\end_inset
-
- se puede expresar de forma única como 
-\begin_inset Formula $\sum_{i\in I}x_{i}$
-\end_inset
-
- con cada 
-\begin_inset Formula $x_{i}\in H_{i}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $H$
-\end_inset
-
- es un 
-\begin_inset Formula $\mathbb{K}$
-\end_inset
-
--espacio de Hilbert y 
-\begin_inset Formula $(H_{i})_{i\in I}$
-\end_inset
-
- es una familia de subespacios cerrados de 
-\begin_inset Formula $H$
-\end_inset
-
- mutuamente ortogonales con 
-\begin_inset Formula $H=\overline{\text{span}\{H_{i}\}_{i\in I}}$
-\end_inset
-
-, entonces 
-\begin_inset Formula $H$
-\end_inset
-
- es isométricamente isomorfo a 
-\begin_inset Formula $\bigoplus_{i\in I}H_{i}$
-\end_inset
-
-, e identificamos 
-\begin_inset Formula $H$
-\end_inset
-
- con 
-\begin_inset Formula $\bigoplus_{i\in I}H_{i}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Desigualdad de Bessel:
-\series default
- Sean 
-\begin_inset Formula $H$
-\end_inset
-
- un espacio prehilbertiano y 
-\begin_inset Formula $\{e_{i}\}_{i\in I}\subseteq H$
-\end_inset
-
- una familia ortonormal, para 
-\begin_inset Formula $x\in H$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}\leq\Vert x\Vert^{2}.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Para un conjunto 
-\begin_inset Formula $I$
-\end_inset
-
- arbitrario, llamamos 
-\begin_inset Formula $\ell^{2}(I)\coloneqq\bigoplus_{i\in I}\mathbb{K}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de la base hilbertiana:
-\series default
- Sean 
-\begin_inset Formula $H$
-\end_inset
-
- un espacio de Hilbert y 
-\begin_inset Formula $\{e_{i}\}_{i\in I}\subseteq H$
-\end_inset
-
- una familia ortonormal, 
-\begin_inset Formula $\{e_{i}\}_{i\in I}$
-\end_inset
-
- es ortonormal maximal (por inclusión) si y sólo si 
-\begin_inset Formula $\forall x\in H,(\forall i\in I,\langle x,e_{i}\rangle=0\implies x=0)$
-\end_inset
-
-, si y sólo si es un conjunto total, si y sólo si 
-\begin_inset Formula $\hat{}:H\to\ell^{2}(I)$
-\end_inset
-
- dada por 
-\begin_inset Formula $\hat{x}\coloneqq(\langle x,e_{i}\rangle)_{i\in I}$
-\end_inset
-
- es inyectiva, si y sólo si todo 
-\begin_inset Formula $x\in H$
-\end_inset
-
- admite un 
-\series bold
-desarrollo de Fourier
-\series default
- 
-\begin_inset Formula $x=\sum_{i\in I}\langle x,e_{i}\rangle e_{i}$
-\end_inset
-
-, si y sólo si 
-\begin_inset Formula $\forall x,y\in H,\langle x,y\rangle=\sum_{i\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{i}\rangle}$
-\end_inset
-
-, si y sólo si todo 
-\begin_inset Formula $x\in H$
-\end_inset
-
- cumple la 
-\series bold
-identidad de Parseval
-\series default
-, 
-\begin_inset Formula $\Vert x\Vert^{2}=\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}$
-\end_inset
-
-, y entonces decimos que 
-\begin_inset Formula $(e_{i})_{i\in I}$
-\end_inset
-
- es una 
-\series bold
-base hilbertiana
-\series default
- de 
-\begin_inset Formula $H$
-\end_inset
-
- o un 
-\series bold
-sistema ortonormal completo
-\series default
-.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $1\implies2]$
-\end_inset
-
- Entonces 
-\begin_inset Formula $x\bot\{e_{i}\}_{i\in I}$
-\end_inset
-
-, por lo que si 
-\begin_inset Formula $x\neq0$
-\end_inset
-
-, 
-\begin_inset Formula $\{e_{i}\}_{i\in I}\cup\{x\}$
-\end_inset
-
- sería ortogonal.
-\begin_inset Formula $\#$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $2\iff3]$
-\end_inset
-
- Sabemos que un 
-\begin_inset Formula $S\subseteq H$
-\end_inset
-
- es total si y sólo si 
-\begin_inset Formula $S^{\bot}=0$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $2\iff4]$
-\end_inset
-
- Por ser 
-\begin_inset Formula $\hat{}$
-\end_inset
-
- lineal.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $4\implies5]$
-\end_inset
-
- 
-\begin_inset Formula $\widehat{\sum_{i}\langle x,e_{i}\rangle e_{i}}=\sum_{i}\langle x,e_{i}\rangle\hat{e}_{i}=\sum_{i}\langle x,e_{i}\rangle e_{i}=\hat{x}$
-\end_inset
-
-, y por inyectividad 
-\begin_inset Formula $x=\sum_{i\in I}\langle x,e_{i}\rangle e_{i}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $5\implies6]$
-\end_inset
-
- 
-\begin_inset Formula $\langle x,y\rangle=\sum_{i,j\in I}\langle\langle x,e_{i}\rangle e_{i},\langle y,e_{j}\rangle e_{j}\rangle=\sum_{i,j\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{j}\rangle}\langle e_{i},e_{j}\rangle=\sum_{i\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{j}\rangle}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $6\implies7]$
-\end_inset
-
- Basta tomar 
-\begin_inset Formula $x=y$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $7\implies1]$
-\end_inset
-
- Si fuera 
-\begin_inset Formula $\{e_{i}\}_{i}\subsetneq M\subseteq H$
-\end_inset
-
- con 
-\begin_inset Formula $M$
-\end_inset
-
- ortonormal, para 
-\begin_inset Formula $x\in M\setminus\{e_{i}\}_{i}$
-\end_inset
-
-, 
-\begin_inset Formula $1=\Vert x\Vert^{2}=\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}=0\#$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Primer teorema de Riesz-Fischer:
-\series default
- Si 
-\begin_inset Formula $H$
-\end_inset
-
- es un espacio prehilbertiano con una familia ortonormal 
-\begin_inset Formula $\{e_{i}\}_{i\in I}$
-\end_inset
-
- y 
-\begin_inset Formula $\hat{}:H\to\mathbb{K}^{I}$
-\end_inset
-
- viene dada por 
-\begin_inset Formula $\hat{x}\coloneqq(\langle x,e_{i}\rangle)_{i\in I}$
-\end_inset
-
-, 
-\begin_inset Formula $\hat{}$
-\end_inset
-
- es lineal y continua con imagen contenida en 
-\begin_inset Formula $\ell^{2}(I)$
-\end_inset
-
- e igual a 
-\begin_inset Formula $\ell^{2}(I)$
-\end_inset
-
- si 
-\begin_inset Formula $H$
-\end_inset
-
- es de Hilbert.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $H$
-\end_inset
-
- es un espacio de Hilbert, todo espacio ortonormal de vectores en 
-\begin_inset Formula $H$
-\end_inset
-
- se puede completar a una base hilbertiana de 
-\begin_inset Formula $H$
-\end_inset
-
-, y en particular todo espacio de Hilbert posee una base hilbertiana y es
- isométricamente isomorfo a un 
-\begin_inset Formula $\ell^{2}(I)$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Los espacios de Hilbert 
-\begin_inset Formula $\ell^{2}(I)$
-\end_inset
-
- y 
-\begin_inset Formula $\ell^{2}(J)$
-\end_inset
-
- son topológicamente isomorfos si y sólo si 
-\begin_inset Formula $|I|=|J|$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\series bold
-dimensión hilbertiana
-\series default
- de un espacio de Hilbert al cardinal de cualquier base hilbertiana.
- 
-\series bold
-Segundo teorema de Riesz-Fischer:
-\series default
- Si 
-\begin_inset Formula $H$
-\end_inset
-
- es de dimensión infinita, 
-\begin_inset Formula $\dim H=\aleph_{0}\coloneqq|\mathbb{N}|$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $H\cong\ell^{2}$
-\end_inset
-
-, si y sólo si 
-\begin_inset Formula $H$
-\end_inset
-
- es separable.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $1\iff2]$
-\end_inset
-
- Por lo anterior.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $2\implies3]$
-\end_inset
-
- Visto.
-\end_layout
-
-\begin_layout Description
-\begin_inset Formula $3\implies2]$
-\end_inset
-
- Dado 
-\begin_inset Formula $\{x_{n}\}_{n\in\mathbb{N}}\subseteq H$
-\end_inset
-
- denso, como 
-\begin_inset Formula $H$
-\end_inset
-
- es de dimensión infinita, existe una subsucesión 
-\begin_inset Formula $(x_{n_{k}})_{k}$
-\end_inset
-
- linealmente independiente de 
-\begin_inset Formula $(x_{n})_{n}$
-\end_inset
-
- con 
-\begin_inset Formula $\text{span}\{x_{n}\}_{n}=\text{span}\{x_{n_{k}}\}_{k}$
-\end_inset
-
-, luego 
-\begin_inset Formula $\overline{\text{span}\{x_{n_{k}}\}_{k}}=H$
-\end_inset
-
- y el proceso de ortonormalización de Gram-Schmidt nos da una base hilbertiana
- numerable de 
-\begin_inset Formula $H$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Así, si 
-\begin_inset Formula $Z\leq_{\mathbb{K}}\ell^{2}$
-\end_inset
-
- es cerrado de dimensión infinita, 
-\begin_inset Formula $Z\cong\ell^{2}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Section
-Aproximaciones por polinomios
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $I\subseteq\mathbb{R}$
-\end_inset
-
- es un intervalo cerrado, llamamos 
-\begin_inset Formula ${\cal C}(I)$
-\end_inset
-
- al conjunto de funciones 
-\begin_inset Formula $I\to\mathbb{R}$
-\end_inset
-
- continuas en el interior de 
-\begin_inset Formula $I$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Korovkin:
-\series default
- Sean 
-\begin_inset Formula $p_{0},p_{1},p_{2}:[a,b]\subseteq\mathbb{R}\to\mathbb{R}$
-\end_inset
-
- dadas por 
-\begin_inset Formula $p_{k}(t)\coloneqq t^{k}$
-\end_inset
-
- y 
-\begin_inset Formula $(P_{n}:{\cal C}([a,b])\to{\cal C}([a,b]))_{n}$
-\end_inset
-
- una sucesión de funciones lineales positivas (
-\begin_inset Formula $\forall f\in{\cal C}([a,b]),(f\geq0\implies P_{n}(f)\geq0)$
-\end_inset
-
-) con 
-\begin_inset Formula $\lim_{n}\Vert P_{n}(p_{k})-p_{k}\Vert_{\infty}=0$
-\end_inset
-
- para 
-\begin_inset Formula $k\in\{0,1,2\}$
-\end_inset
-
-, entonces, para 
-\begin_inset Formula $f\in{\cal C}([a,b])$
-\end_inset
-
-, 
-\begin_inset Formula $\lim_{n}\Vert P_{n}(f)-f\Vert_{\infty}=0$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Weierstrass:
-\series default
- El conjunto de polinomios en una variable es denso 
-\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Así, para 
-\begin_inset Formula $f\in{\cal C}([a,b])$
-\end_inset
-
-, se puede encontrar una sucesión de polinomios que converja uniformemente
- a 
-\begin_inset Formula $f$
-\end_inset
-
-.
- Hacerlo con polinomios de interpolación por nodos prefijados no es una
- buena estrategia ya que para toda secuencia de nodos de interpolación en
- 
-\begin_inset Formula $[a,b]$
-\end_inset
-
-, existe 
-\begin_inset Formula $f\in{\cal C}([a,b])$
-\end_inset
-
- para la que los polinomios de interpolación en dichos nodos no converge
- uniformemente a 
-\begin_inset Formula $f$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
- Si se hace con nodos equidistantes se da el fenómeno de Runge.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Čebyšev:
-\series default
- Para 
-\begin_inset Formula $f\in{\cal C}([a,b])$
-\end_inset
-
- y 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
-, si 
-\begin_inset Formula $K_{n}\subseteq\mathbb{K}[X]$
-\end_inset
-
- es el conjunto de polinomio de grado máximo 
-\begin_inset Formula $n$
-\end_inset
-
-, 
-\begin_inset Formula $p:K_{n}\mapsto\Vert f-p\Vert_{\infty}$
-\end_inset
-
- tiene un único mínimo 
-\begin_inset Formula $p_{n}$
-\end_inset
-
-, y 
-\begin_inset Formula $(p_{n})_{n}$
-\end_inset
-
- converge uniformemente a 
-\begin_inset Formula $f$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Un 
-\series bold
-polinomio trigonométrico real
-\series default
- es una función 
-\begin_inset Formula $p:\mathbb{R}\to\mathbb{R}$
-\end_inset
-
- de la forma
-\begin_inset Formula 
-\[
-p(x)\coloneqq\sum_{n=0}^{m}(a_{n}\cos(nx)+b_{n}\sin(nx))
-\]
-
-\end_inset
-
-para ciertos 
-\begin_inset Formula $a_{n},b_{n}\in\mathbb{R}$
-\end_inset
-
-.
- 
-\series bold
-Teorema de Weierstrass:
-\series default
- Si 
-\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{R}$
-\end_inset
-
- es continua con 
-\begin_inset Formula $f(-\pi)=f(\pi)$
-\end_inset
-
-, para cada 
-\begin_inset Formula $\varepsilon>0$
-\end_inset
-
- existe un polinomio trigonométrico real 
-\begin_inset Formula $p$
-\end_inset
-
- con 
-\begin_inset Formula $\Vert f-p\Vert_{\infty}<\varepsilon$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Para 
-\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{C}$
-\end_inset
-
- integrable y 
-\begin_inset Formula $r\in\mathbb{Z}$
-\end_inset
-
-, llamamos 
-\series bold
-
-\begin_inset Formula $r$
-\end_inset
-
--ésimo coeficiente de Fourier
-\series default
- de 
-\begin_inset Formula $f$
-\end_inset
-
- a
-\begin_inset Formula 
-\[
-\hat{f}(r)\coloneqq\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)\text{e}^{-\text{i}rt}\dif t,
-\]
-
-\end_inset
-
-y 
-\series bold
-serie de Fourier
-\series default
- de 
-\begin_inset Formula $f$
-\end_inset
-
- a la serie formal
-\begin_inset Formula 
-\[
-\sum_{r\in\mathbb{Z}}\hat{f}(r)\text{e}^{-\text{i}rt}.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Para 
-\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{R}$
-\end_inset
-
- integrable y 
-\begin_inset Formula $n\in\mathbb{N}^{*}$
-\end_inset
-
-, llamando
-\begin_inset Formula 
-\begin{align*}
-a_{0} & \coloneqq\frac{1}{2\pi}\int_{-\pi}^{\pi}f, & a_{n} & \coloneqq\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(nt)\dif t, & b_{n} & \coloneqq\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\sin(nt)\dif t,
-\end{align*}
-
-\end_inset
-
-la 
-\series bold
-serie de Fourier real
-\series default
- de 
-\begin_inset Formula $f$
-\end_inset
-
- es
-\begin_inset Formula 
-\[
-\sum_{n=0}^{\infty}a_{n}\cos(nt)+\sum_{n=1}^{\infty}b_{n}\sin(nt).
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Como 
-\series bold
-teorema
-\series default
-, sean 
-\begin_inset Formula $([-\pi,\pi],\Sigma,\mu)$
-\end_inset
-
- es el espacio de medida usual en 
-\begin_inset Formula $[-\pi,\pi]$
-\end_inset
-
-, 
-\begin_inset Formula $M_{\mathbb{R}}\coloneqq L_{\mathbb{R}}^{2}([-\pi,\pi],\Sigma,\frac{\mu}{\pi})$
-\end_inset
-
- y 
-\begin_inset Formula $M_{\mathbb{C}}\coloneqq L_{\mathbb{C}}^{2}([-\pi,\pi],\Sigma,\frac{\mu}{2\pi})$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-El 
-\series bold
-sistema trigonométrico
-\series default
- 
-\begin_inset Formula $(\text{e}^{\text{i}rt})_{r\in\mathbb{Z}}$
-\end_inset
-
- es una base hilbertiana de 
-\begin_inset Formula $M_{\mathbb{C}}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $(\cos(nt))_{n\in\mathbb{N}}\star(\sin(nt))_{n\in\mathbb{N}^{*}}$
-\end_inset
-
- es una base hilbertiana de 
-\begin_inset Formula $M_{\mathbb{R}}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $f\in M_{\mathbb{C}}$
-\end_inset
-
-, 
-\begin_inset Formula $f$
-\end_inset
-
- coincide con su serie de Fourier en 
-\begin_inset Formula $\Vert\cdot\Vert_{2}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $f\in M_{\mathbb{R}}$
-\end_inset
-
-, 
-\begin_inset Formula $f$
-\end_inset
-
- coincide con su serie de Fourier real en 
-\begin_inset Formula $\Vert\cdot\Vert_{2}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula ${\cal F}:M_{\mathbb{C}}\to\ell^{2}(\mathbb{Z})$
-\end_inset
-
- que asigna a cada función su familia de coeficientes de Fourier 
-\begin_inset Formula $(\hat{f}(n))_{n\in\mathbb{Z}}$
-\end_inset
-
- es un isomorfismo de espacios de Hilbert.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Un 
-\series bold
-peso
-\series default
- en un intervalo cerrado 
-\begin_inset Formula $I\subseteq\mathbb{R}$
-\end_inset
-
- es una 
-\begin_inset Formula $p\in{\cal C}(I)$
-\end_inset
-
- estrictamente positiva tal que
-\begin_inset Formula 
-\[
-\forall n\in\mathbb{N},\int_{I}|t|^{n}p(t)\dif t<\infty.
-\]
-
-\end_inset
-
-Entonces 
-\begin_inset Formula $\langle\cdot,\cdot\rangle:{\cal C}(I)\times{\cal C}(I)\to[-\infty,+\infty]$
-\end_inset
-
- dada por
-\begin_inset Formula 
-\[
-\langle f,g\rangle\coloneqq\int_{I}f\overline{g}p
-\]
-
-\end_inset
-
-es un producto escalar en 
-\begin_inset Formula $H_{p}\coloneqq\{f\in{\cal C}(I)\mid\langle f,f\rangle<\infty\}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\series bold
-sucesión de polinomios ortonormales
-\series default
- asociada a 
-\begin_inset Formula $\langle\cdot,\cdot\rangle$
-\end_inset
-
- o al peso 
-\begin_inset Formula $p$
-\end_inset
-
- en 
-\begin_inset Formula $I$
-\end_inset
-
- a una sucesión 
-\begin_inset Formula $\{P_{n}\}_{n\in\mathbb{N}}\subseteq H_{p}$
-\end_inset
-
- de polinomios con 
-\begin_inset Formula $\text{span}\{1,t,\dots,t^{n}\}=\text{span}\{P_{0},P_{1},\dots,P_{n}\}$
-\end_inset
-
- para cada 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
-, y entonces, para 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $P_{n}$
-\end_inset
-
- es un polinomio de grado 
-\begin_inset Formula $n$
-\end_inset
-
- con coeficientes reales.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $P_{n}$
-\end_inset
-
- es ortogonal en 
-\begin_inset Formula $H_{p}$
-\end_inset
-
- al subespacio de polinomios de grado menor que 
-\begin_inset Formula $n$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $P_{n}$
-\end_inset
-
- tiene 
-\begin_inset Formula $n$
-\end_inset
-
- raíces distintas en 
-\begin_inset Formula $(a,b)$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Ejemplos:
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Polinomios de Legendre.
-\series default
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula 
-\begin{align*}
-I & =[-1,1], & p(t) & =1, & P_{n}(t) & =\frac{\sqrt{\frac{2n+1}{2}}}{2^{n}n!}\od[n]{(t^{2}-1)^{n}}{t}.
-\end{align*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Polinomios de Laguerre.
-\series default
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\series bold
-
-\begin_inset Formula 
-\begin{align*}
-I & =[0,\infty), & p(t) & =\text{e}^{-t}, & P_{n}(t) & =\frac{\text{e}^{t}}{n!}\od[n]{\text{e}^{-t}t^{n}}{t}.
-\end{align*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Polinomios de Hermite.
-\series default
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula 
-\begin{align*}
-I & =(-\infty,\infty), & p(t) & =\text{e}^{-t^{2}}, & P_{n}(t) & =\frac{\text{e}^{t^{2}}}{\sqrt[4]{\pi}\sqrt{2^{n}n!}}\od[n]{\text{e}^{-t^{2}}}{t}.
-\end{align*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-
-\series bold
-Polinomios de Čebyšev.
-\series default
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula 
-\begin{align*}
-I & =[-1,1], & p(t) & =\frac{1}{\sqrt{1-t^{2}}}, & P_{n}(t) & =\cos(n\arccos t),
-\end{align*}
-
-\end_inset
-
-siendo 
-\begin_inset Formula $\arccos:[-1,1]\to[0,\pi]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Una sucesión de polinomios ortonormales asociada a un peso 
-\begin_inset Formula $p$
-\end_inset
-
- en un intervalo compacto es total en 
-\begin_inset Formula $H_{p}$
-\end_inset
-
-, y en particular los polinomios de Legendre forman una base hilbertiana
- en 
-\begin_inset Formula $L^{2}([-1,1]).$
-\end_inset
-
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $p$
-\end_inset
-
- es un peso en 
-\begin_inset Formula $[a,b]$
-\end_inset
-
- y 
-\begin_inset Formula $a\leq t_{1}<\dots<t_{n}\leq b$
-\end_inset
-
-, se tiene una 
-\series bold
-fórmula de cuadratura gaussiana
-\series default
-, 
-\begin_inset Formula 
-\[
-\int_{a}^{b}fp\approx\sum_{k=1}^{n}A_{k}f(t_{k})
-\]
-
-\end_inset
-
-para ciertos 
-\begin_inset Formula $A_{1},\dots,A_{n}\in\mathbb{R}$
-\end_inset
-
-, y se alcanza la igualdad si 
-\begin_inset Formula $f$
-\end_inset
-
- es un polinomio de grado menor que 
-\begin_inset Formula $n$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Gauss:
-\series default
- Dados un peso 
-\begin_inset Formula $p$
-\end_inset
-
- en 
-\begin_inset Formula $[a,b]$
-\end_inset
-
- con una sucesión de polinomios ortonormales 
-\begin_inset Formula $(P_{n})_{n}$
-\end_inset
-
-, 
-\begin_inset Formula $n\in\mathbb{N}^{*}$
-\end_inset
-
-, 
-\begin_inset Formula $a<t_{1}<\dots<t_{n}<b$
-\end_inset
-
- y 
-\begin_inset Formula $A_{1},\dots,A_{n}\in\mathbb{R}$
-\end_inset
-
-, si
-\begin_inset Formula 
-\[
-\int_{a}^{b}fp=\sum_{k=1}^{n}A_{k}f(t_{k})
-\]
-
-\end_inset
-
-para todo polinomio 
-\begin_inset Formula $f$
-\end_inset
-
- de grado menor que 
-\begin_inset Formula $n$
-\end_inset
-
-, esta fórmula se para polinomios de grado menor que 
-\begin_inset Formula $2n$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $t_{1},\dots,t_{n}$
-\end_inset
-
- son los ceros de 
-\begin_inset Formula $P_{n}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Stieltjes:
-\series default
- Sean 
-\begin_inset Formula $p$
-\end_inset
-
- un peso en 
-\begin_inset Formula $[a,b]$
-\end_inset
-
- con una sucesión de polinomios ortonormales 
-\begin_inset Formula $(P_{n})_{n}$
-\end_inset
-
- y, para 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
-, 
-\begin_inset Formula $t_{n1}<\dots<t_{nn}$
-\end_inset
-
- los ceros de 
-\begin_inset Formula $P_{n}$
-\end_inset
-
- y 
-\begin_inset Formula $A_{n1},\dots,A_{nn}\in\mathbb{R}$
-\end_inset
-
- los correspondientes coeficientes en la fórmula de cuadratura gaussiana,
- para 
-\begin_inset Formula $f\in{\cal C}([a,b])$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\int_{a}^{b}fp=\lim_{n}\sum_{k=1}^{n}A_{nk}f(t_{nk}).
-\]
-
-\end_inset
-
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-El espacio de Bergman
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\begin_inset Formula $D(a,r)\coloneqq B(a,r)\subseteq\mathbb{C}$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $\Omega\subseteq\mathbb{C}$
-\end_inset
-
- es abierto, 
-\begin_inset Formula ${\cal H}(\Omega)$
-\end_inset
-
- es el conjunto de las funciones holomorfas en 
-\begin_inset Formula $\Omega$
-\end_inset
-
-, y para 
-\begin_inset Formula $f\in{\cal H}(\Omega)$
-\end_inset
-
- y 
-\begin_inset Formula $\overline{D(a,r)}\subseteq\Omega$
-\end_inset
-
-, la serie 
-\begin_inset Formula $\sum_{n\in\mathbb{N}}a_{n}(z-a)^{n}$
-\end_inset
-
- con 
-\begin_inset Formula $z\in D(a,r)$
-\end_inset
-
- converge uniformemente a 
-\begin_inset Formula $f$
-\end_inset
-
- en compactos de 
-\begin_inset Formula $D(a,r)$
-\end_inset
-
- para ciertos 
-\begin_inset Formula $a_{n}\in\mathbb{C}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $\Omega\subseteq\mathbb{C}$
-\end_inset
-
- es abierto, llamamos 
-\begin_inset Formula ${\cal T}_{\text{K}}$
-\end_inset
-
- a la topología en 
-\begin_inset Formula ${\cal H}(\Omega)$
-\end_inset
-
- de convergencia uniforme sobre compactos, y 
-\series bold
-espacio de Bergman
-\series default
- en el abierto 
-\begin_inset Formula $\Omega\subseteq\mathbb{C}$
-\end_inset
-
- a
-\begin_inset Formula 
-\[
-A^{2}(\Omega)\coloneqq\left\{ f\in{\cal H}(\Omega)\;\middle|\;\int_{\Omega}|f|^{2}<\infty\right\} ,
-\]
-
-\end_inset
-
-un subespacio cerrado y separable de 
-\begin_inset Formula $L^{2}(\Omega)$
-\end_inset
-
- que es pues un espacio de Hilbert numerable con 
-\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$
-\end_inset
-
-, y en el que la topología inducida por 
-\begin_inset Formula $L^{2}(\Omega)$
-\end_inset
-
- es más fina que la inducida por 
-\begin_inset Formula ${\cal T}_{\text{K}}$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $\Omega\subseteq\mathbb{C}$
-\end_inset
-
- es abierto, 
-\begin_inset Formula $(\omega_{n})_{n}$
-\end_inset
-
- es base hilbertiana de 
-\begin_inset Formula $A^{2}(\Omega)$
-\end_inset
-
- y 
-\begin_inset Formula $f\in A^{2}(\Omega)$
-\end_inset
-
-, el desarrollo en serie de Fourier de 
-\begin_inset Formula $f$
-\end_inset
-
-, 
-\begin_inset Formula $\sum_{n}\langle f,\omega_{n}\rangle\omega_{n}$
-\end_inset
-
-, converge uniformemente a 
-\begin_inset Formula $f$
-\end_inset
-
- en compactos de 
-\begin_inset Formula $\Omega$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $\psi_{n}(z)\coloneqq(z-a)^{n}$
-\end_inset
-
-, 
-\begin_inset Formula $(\frac{\psi_{n}}{\Vert\psi_{n}\Vert})_{n}$
-\end_inset
-
- es una base hilbertiana de 
-\begin_inset Formula $A^{2}(D(a,r))$
-\end_inset
-
-, y el desarrollo en serie de potencias es el desarrollo en serie de Fourier
- sobre esta base.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Como 
-\series bold
-teorema
-\series default
-, si 
-\begin_inset Formula $\Omega\subsetneq\mathbb{C}$
-\end_inset
-
- es un abierto simplemente conexo y 
-\begin_inset Formula $f:\Omega\to D(0,1)$
-\end_inset
-
- es un isomorfismo,
-\begin_inset Formula 
-\[
-\left(z\mapsto\sqrt{\frac{n}{\pi}}(f(z))^{n-1}\dot{f}(z)\right)_{n}
-\]
-
-\end_inset
-
-es base hilbertiana de 
-\begin_inset Formula $A^{2}(\Omega)$
-\end_inset
-
-, y en particular para 
-\begin_inset Formula $R>0$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-\left(z\mapsto\sqrt{\frac{n}{\pi}}R^{-n}z^{n-1}\right)_{n}
-\]
-
-\end_inset
-
- es base hilbertiana de 
-\begin_inset Formula $A^{2}(D(0,R))$
-\end_inset
-
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
diff --git a/af/n2.lyx b/af/n2.lyx
index e043d8a..fc349c6 100644
--- a/af/n2.lyx
+++ b/af/n2.lyx
@@ -82,98 +82,139 @@
 \begin_body
 
 \begin_layout Standard
-Algunos operadores acotados en espacios de Hilbert:
+David Hilbert (1862–1943) fue un influyente matemático alemán que formuló
+ la teoría de los espacios de Hilbert.
+ En 1900 publicó una lista de 23 problemas que marcarían en buena medida
+ el progreso matemático en el siglo XX, y presentó 10 de ellos en el 
+\emph on
+\lang english
+International Congress of Mathematicians
+\emph default
+\lang spanish
+ de París de 1900.
+ Fue editor jefe de 
+\emph on
+\lang ngerman
+Mathematische Annalen
+\emph default
+\lang spanish
+, una revista matemática muy prestigiosa por casi 150 años, y tuvo discípulos
+ como 
+\lang ngerman
+Alfréd Haar, Erhard Schmidt, Hugo Steihaus, Hermann Weyl o Ernst Zermelo
+\lang spanish
+.
 \end_layout
 
-\begin_layout Enumerate
-Sean 
-\begin_inset Formula $G$
+\begin_layout Standard
+Dado un 
+\begin_inset Formula $\mathbb{K}$
 \end_inset
 
- y 
+-espacio vectorial 
 \begin_inset Formula $H$
 \end_inset
 
- espacios prehilbertianos y 
-\begin_inset Formula $G$
+, 
+\begin_inset Formula $\langle\cdot,\cdot\rangle:H\times H\to\mathbb{K}$
 \end_inset
 
- de dimensión finita con base 
-\begin_inset Formula $(e_{i})_{i}$
+ es una 
+\series bold
+forma hermitiana
+\series default
+ si para 
+\begin_inset Formula $a,b\in\mathbb{K}$
 \end_inset
 
-, todo homomorfismo 
-\begin_inset Formula $T:G\to H$
+ y 
+\begin_inset Formula $x,y,z\in H$
 \end_inset
 
- es acotado con
-\begin_inset Formula 
-\[
-\Vert T\Vert\leq\sqrt{\sum_{i}\Vert Te_{i}\Vert^{2}}.
-\]
+ se tiene 
+\begin_inset Formula $\langle ax+by,z\rangle=a\langle x,z\rangle+b\langle y,z\rangle$
+\end_inset
 
+ y 
+\begin_inset Formula $\langle x,y\rangle=\overline{\langle y,x\rangle}$
 \end_inset
 
+, y es 
+\series bold
+definida positiva
+\series default
+ si para 
+\begin_inset Formula $x\in H\setminus0$
+\end_inset
 
-\begin_inset Note Note
-status open
+ es 
+\begin_inset Formula $\langle x,x\rangle\in\mathbb{R}^{+}$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
+.
+ Un 
+\series bold
+producto escalar
+\series default
+ es una forma hermitiana definida positiva, y un 
+\series bold
+espacio prehilbertiano
+\series default
+ es par formado por un espacio vectorial y un producto escalar sobre este.
 \end_layout
 
+\begin_layout Standard
+Dado un espacio prehilbertiano 
+\begin_inset Formula $(H,\langle\cdot,\cdot\rangle)$
 \end_inset
 
-
+:
 \end_layout
 
 \begin_layout Enumerate
-Sean 
-\begin_inset Formula $G$
-\end_inset
-
- y 
-\begin_inset Formula $H$
-\end_inset
 
+\series bold
+Desigualdad de Cauchy-Schwartz:
+\series default
  
-\begin_inset Formula $\mathbb{K}$
+\begin_inset Formula $\forall x,y\in H,|\langle x,y\rangle|^{2}\leq\langle x,x\rangle\langle y,y\rangle$
 \end_inset
 
--espacios de Hilbert de dimensión 
-\begin_inset Formula $\aleph_{0}$
+, con igualdad si y sólo si 
+\begin_inset Formula $x$
 \end_inset
 
- con bases ortonormales 
-\begin_inset Formula $(e_{n})_{n}$
+ e 
+\begin_inset Formula $y$
 \end_inset
 
- y 
-\begin_inset Formula $(f_{n})_{n}$
-\end_inset
+ son linealmente dependientes.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
 
- y 
-\begin_inset Formula $\{a_{n}\}_{n}\subseteq\mathbb{K}$
 \end_inset
 
- una sucesión acotada, el 
-\series bold
-operador diagonal
-\series default
- 
-\begin_inset Formula $T:G\to H$
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $H$
 \end_inset
 
- dado por
-\begin_inset Formula 
-\[
-T(x)\coloneqq\sum_{n=1}^{\infty}a_{n}\langle x,e_{n}\rangle f_{n}
-\]
+ es un espacio normado con la norma 
+\begin_inset Formula $\Vert x\Vert\coloneqq\sqrt{\langle x,x\rangle}$
+\end_inset
 
+, y para 
+\begin_inset Formula $x,y\in H$
 \end_inset
 
-es acotado con 
-\begin_inset Formula $\Vert T\Vert=\sup_{n}|a_{n}|$
+, 
+\begin_inset Formula $\Vert x+y\Vert=\Vert x\Vert+\Vert y\Vert\iff x=0\lor y=0\lor\exists a>0:x=ay$
 \end_inset
 
 .
@@ -190,28 +231,60 @@ nproof
 \end_layout
 
 \begin_layout Enumerate
-Si 
-\begin_inset Formula $g\in L^{\infty}([a,b])$
+Para 
+\begin_inset Formula $a,b\in\mathbb{K}$
 \end_inset
 
-, el 
-\series bold
-operador multiplicación por 
-\begin_inset Formula $g$
+ y 
+\begin_inset Formula $x,y,z\in H$
 \end_inset
 
+, 
+\begin_inset Formula $\langle x,ay+bz\rangle=\overline{a}\langle x,y\rangle+\overline{b}\langle x,z\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $x,y\in H$
+\end_inset
 
-\series default
 , 
-\begin_inset Formula $T:L^{2}([a,b])\to L^{2}([a,b])$
+\begin_inset Formula $\Vert x+y\Vert^{2}=\Vert x\Vert^{2}+\Vert y\Vert^{2}+2\text{Re}\langle x,y\rangle$
 \end_inset
 
- dado por 
-\begin_inset Formula $Tf\coloneqq gf$
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\Vert x+y\Vert^{2}=\langle x+y,x+y\rangle=\langle x,x\rangle+\langle x,y\rangle+\overline{\langle x,y\rangle}+\langle y,y\rangle$
 \end_inset
 
-, es acotado con 
-\begin_inset Formula $\Vert T\Vert=\Vert g\Vert_{\infty}$
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+
+\series bold
+Identidades de polarización:
+\series default
+ Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio prehilbertiano y 
+\begin_inset Formula $x,y\in H$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\langle x,y\rangle=\frac{1}{4}(\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2}+\text{i}\Vert x+\text{i}y\Vert^{2}-\text{i}\Vert x-\text{i}y\Vert^{2})$
 \end_inset
 
 .
@@ -228,171 +301,207 @@ nproof
 \end_layout
 
 \begin_layout Enumerate
-Sean 
-\begin_inset Formula $G$
-\end_inset
-
- y 
+Si 
 \begin_inset Formula $H$
 \end_inset
 
- 
-\begin_inset Formula $\mathbb{K}$
+ se define sobre 
+\begin_inset Formula $\mathbb{R}$
 \end_inset
 
--espacios de Hilbert de dimensión 
-\begin_inset Formula $\aleph_{0}$
+, 
+\begin_inset Formula $\langle x,y\rangle=\frac{1}{4}(\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2})$
 \end_inset
 
- con bases ortonormales respectivas 
-\begin_inset Formula $(u_{n})_{n}$
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de von Neumann:
+\series default
+ Un espacio normado 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
 \end_inset
 
- y 
-\begin_inset Formula $(v_{n})_{n}$
+ admite un producto escalar 
+\begin_inset Formula $\langle\cdot,\cdot\rangle$
 \end_inset
 
- y 
-\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+ en 
+\begin_inset Formula $X$
 \end_inset
 
- una matriz infinita con 
-\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<\infty$
+ con 
+\begin_inset Formula $\langle x,x\rangle\equiv\Vert x\Vert^{2}$
 \end_inset
 
-, 
-\begin_inset Formula $T:G\to H$
+ si y sólo si 
+\begin_inset Formula $\Vert\cdot\Vert$
 \end_inset
 
- dado por
+ verifica la 
+\series bold
+ley del paralelogramo:
+\series default
+
 \begin_inset Formula 
 \[
-T(x)\coloneqq\sum_{i,j}a_{ij}\langle x,u_{i}\rangle v_{j}
+\forall x,y\in H,\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2}=2(\Vert x\Vert^{2}+\Vert y\Vert^{2}).
 \]
 
 \end_inset
 
-es un operador acotado con 
-\begin_inset Formula $\Vert T\Vert\leq\sqrt{\sum_{i,j}|a_{ij}|^{2}}$
-\end_inset
 
-.
-\begin_inset Note Note
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
-nproof
-\end_layout
-
+\begin_inset Formula $\implies]$
 \end_inset
 
 
 \end_layout
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
-\end_inset
-
-, el 
-\series bold
-operador integral con núcleo 
-\begin_inset Formula $k$
 \end_inset
 
-
-\series default
-, 
-\begin_inset Formula $K:L^{2}([a,b])\to L^{2}([a,b])$
+En general 
+\begin_inset Formula $\langle x,y+z\rangle=\overline{\langle y+z,x\rangle}=\overline{\langle y,x\rangle}+\overline{\langle z,x\rangle}=\langle x,y\rangle+\langle x,z\rangle$
 \end_inset
 
- dado por
+, de donde
 \begin_inset Formula 
-\[
-K(f)(t)\coloneqq\int_{a}^{b}k(t,s)f(s)\dif s,
-\]
+\begin{multline*}
+\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2}=\langle x+y,x+y\rangle+\langle x-y,x-y\rangle=\\
+=\langle x,x\rangle+\langle x,y\rangle+\langle y,x\rangle+\langle y,y\rangle+\langle x,x\rangle-\langle x,y\rangle-\langle y,x\rangle+\langle y,y\rangle=2(\Vert x\Vert^{2}+\Vert y\Vert^{2}).
+\end{multline*}
 
 \end_inset
 
-es acotado con 
-\begin_inset Formula $\Vert K\Vert\leq\sqrt{\iint_{[a,b]\times[a,b]}|k|^{2}}$
-\end_inset
 
-.
-\begin_inset Note Note
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
-nproof
-\end_layout
-
+\begin_inset Formula $\impliedby]$
 \end_inset
 
 
 \end_layout
 
-\begin_layout Enumerate
-Una matriz infinita 
-\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
 \end_inset
 
- satisface el 
-\series bold
-test de Schur
-\series default
- si existen 
-\begin_inset Formula $C,D\in\mathbb{R}$
+Definimos 
+\begin_inset Formula $\langle\cdot,\cdot\rangle$
 \end_inset
 
- tales que
+ según la identidad de polarización, y queremos ver que es un producto escalar
+ cuya norma es la inicial.
+ Se tiene
 \begin_inset Formula 
 \begin{align*}
-\forall i\in\mathbb{N},\sum_{j}|a_{ij}| & \leq C, & \forall j\in\mathbb{N}, & \sum_{i}|a_{ij}|\leq D.
+\langle x,x\rangle & =\frac{1}{4}\left(\Vert2x\Vert^{2}-\Vert x-x\Vert^{2}+\text{i}\Vert x+\text{i}x\Vert^{2}-\text{i}\Vert x-\text{i}x\Vert^{2}\right)=\\
+ & =\frac{1}{4}\left(4\Vert x\Vert^{2}+\text{i}|1+\text{i}|^{2}\Vert x\Vert^{2}-\text{i}|1-\text{i}|^{2}\Vert x\Vert^{2}\right)=\Vert x\Vert^{2},
 \end{align*}
 
 \end_inset
 
-Entonces, si 
-\begin_inset Formula $G$
-\end_inset
+y
+\begin_inset Formula 
+\begin{align*}
+4\langle x,y\rangle & =\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2}+\text{i}\Vert x+\text{i}y\Vert^{2}-\text{i}\Vert x-\text{i}y\Vert^{2}\\
+ & =\Vert y+x\Vert^{2}-\Vert y-x\Vert^{2}+\text{i}\Vert y-\text{i}x\Vert-\text{i}\Vert y+\text{i}x\Vert^{2}=4\overline{\langle y,x\rangle}\\
+ & =\Vert-x-y\Vert^{2}-\Vert-x+y\Vert^{2}+\text{i}\Vert-x-\text{i}y\Vert^{2}-\text{i}\Vert-x+\text{i}y\Vert^{2}=-4\langle-x,y\rangle\\
+ & =\Vert\text{i}x+\text{i}y\Vert^{2}-\Vert\text{i}x-\text{i}y\Vert^{2}+\text{i}\Vert\text{i}x-y\Vert^{2}-\text{i}\Vert\text{i}x+y\Vert^{2}=4\frac{\langle\text{i}x,y\rangle}{\text{i}}.
+\end{align*}
 
- y 
-\begin_inset Formula $H$
 \end_inset
 
- son 
-\begin_inset Formula $\mathbb{K}$
+Para ver que 
+\begin_inset Formula $\langle x+z,y\rangle=\langle x,y\rangle+\langle z,y\rangle$
 \end_inset
 
--espacios de Hilbert de dimensión 
-\begin_inset Formula $\aleph_{0}$
-\end_inset
+,
+\begin_inset Formula 
+\begin{multline*}
+\Vert x+z+y\Vert^{2}-\Vert x+z-y\Vert^{2}=\left\Vert \left(x+\frac{y}{2}\right)+\left(z+\frac{y}{2}\right)\right\Vert ^{2}-\left\Vert \left(x+\frac{y}{2}\right)-\left(z+\frac{y}{2}\right)\right\Vert ^{2}=\\
+=2\left\Vert x+\frac{y}{2}\right\Vert ^{2}+2\left\Vert z+\frac{y}{2}\right\Vert ^{2}\cancel{-\Vert x-z\Vert^{2}}-2\left\Vert x-\frac{y}{2}\right\Vert ^{2}-2\left\Vert z-\frac{y}{2}\right\Vert ^{2}\cancel{+\Vert x-z\Vert^{2}},
+\end{multline*}
 
- con bases ortonormales respectivas 
-\begin_inset Formula $(u_{n})_{n}$
 \end_inset
 
- y 
-\begin_inset Formula $(v_{n})_{n}$
-\end_inset
+de donde
+\begin_inset Formula 
+\begin{eqnarray*}
+4\langle x+z,y\rangle & = & \Vert x+z+y\Vert^{2}-\Vert x+z-y\Vert^{2}+\text{i}\Vert x+z+\text{i}y\Vert^{2}-\text{i}\Vert x+z-\text{i}y\Vert^{2}\\
+ & = & 2\left(\left\Vert x+\frac{y}{2}\right\Vert ^{2}+\left\Vert z+\frac{y}{2}\right\Vert ^{2}-\left\Vert x-\frac{y}{2}\right\Vert ^{2}-\left\Vert z-\frac{y}{2}\right\Vert \right)\\
+ &  & +2\text{i}\left(\left\Vert x+\text{i}\frac{y}{2}\right\Vert ^{2}+\left\Vert z+\text{i}\frac{z}{2}\right\Vert ^{2}-\left\Vert x-\text{i}\frac{y}{2}\right\Vert ^{2}-\left\Vert z-\text{i}\frac{y}{2}\right\Vert ^{2}\right)\\
+ & = & 8\left\langle x,\frac{y}{2}\right\rangle +8\left\langle z,\frac{y}{2}\right\rangle ,
+\end{eqnarray*}
 
-, 
-\begin_inset Formula $T:G\to H$
 \end_inset
 
- dada por
+y por tanto
 \begin_inset Formula 
 \[
-T(x)\coloneqq\sum_{i,j}a_{ij}\langle x,u_{i}\rangle v_{j}
+\langle x+z,y\rangle=2\left\langle x,\frac{y}{2}\right\rangle +2\left\langle z,\frac{y}{2}\right\rangle =\langle x,y\rangle+\langle z,y\rangle,
 \]
 
 \end_inset
 
-es un operador acotado con 
-\begin_inset Formula $\Vert T\Vert\leq\sqrt{CD}$
+donde en la segunda igualdad hemos usado la primera igualdad con 
+\begin_inset Formula $z=0$
+\end_inset
+
+ o 
+\begin_inset Formula $x=0$
 \end_inset
 
 .
+ Usando esto y que 
+\begin_inset Formula $\langle-x,y\rangle$
+\end_inset
+
+ es fácil ver que 
+\begin_inset Formula $\langle ax,y\rangle=a\langle x,y\rangle$
+\end_inset
+
+ para 
+\begin_inset Formula $a\in\mathbb{Q}$
+\end_inset
+
+; para 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+ se usa la continuidad de la norma y por tanto del producto escalar, y para
+ 
+\begin_inset Formula $a\in\mathbb{C}$
+\end_inset
+
+ se usa 
+\begin_inset Formula $\langle\text{i}x,y\rangle=\text{i}\langle x,y\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $(\ell^{\infty},\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ y 
+\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{1})$
+\end_inset
+
+ son espacios normados no prehilbertianos.
 \begin_inset Note Note
 status open
 
@@ -405,40 +514,32 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-Sean 
-\begin_inset Formula $k:[a,b]\times[a,b]\to\mathbb{K}$
-\end_inset
-
- medible y 
-\begin_inset Formula $C,D\in\mathbb{R}$
+\begin_layout Standard
+Dos espacios prehilbertianos 
+\begin_inset Formula $(H_{1},\langle\cdot,\cdot\rangle_{1})$
 \end_inset
 
- tales que
-\begin_inset Formula 
-\begin{align*}
-\forall t\in[a,b],\int_{a}^{b}|k(t,s)|\dif s & \leq C, & \forall s\in[a,b], & \int_{a}^{b}|k(t,s)|\dif t\leq D,
-\end{align*}
-
+ y 
+\begin_inset Formula $(H_{2},\langle\cdot,\cdot\rangle_{2})$
 \end_inset
 
-entonces 
-\begin_inset Formula $K:L^{2}([a,b])\to L^{2}([a,b])$
+ son 
+\series bold
+equivalentes
+\series default
+ si existe un isomorfismo algebraico 
+\begin_inset Formula $T:H_{1}\to H_{2}$
 \end_inset
 
- dada por
-\begin_inset Formula 
-\[
-K(f)(t)\coloneqq\int_{a}^{b}k(t,s)f(s)\dif s
-\]
-
+ con 
+\begin_inset Formula $\langle x,y\rangle_{1}=\langle T(x),T(y)\rangle_{2}$
 \end_inset
 
-es un operador acotado con 
-\begin_inset Formula $\Vert K\Vert\leq\sqrt{CD}$
+ para todo 
+\begin_inset Formula $x,y\in H_{1}$
 \end_inset
 
-.
+, si y sólo si existe un isomorfismo isométrico entre los espacios normados.
 \begin_inset Note Note
 status open
 
@@ -456,91 +557,105 @@ Si
 \begin_inset Formula $H$
 \end_inset
 
- es un espacio de Hilbert de dimensión 
-\begin_inset Formula $\aleph_{0}$
+ es un espacio prehilbertiano, 
+\begin_inset Formula $x,y\in H$
 \end_inset
 
- con base ortonormal 
-\begin_inset Formula $(e_{n})_{n}$
+ son 
+\series bold
+ortogonales
+\series default
+, 
+\begin_inset Formula $x\bot y$
 \end_inset
 
-, para 
-\begin_inset Formula $T\in L(H)$
+, si 
+\begin_inset Formula $\langle x,y\rangle=0$
 \end_inset
 
- y 
+.
+ Decimos que 
 \begin_inset Formula $x\in H$
 \end_inset
 
-,
-\begin_inset Formula 
-\[
-T(x)=\sum_{i,j}\langle x,e_{j}\rangle\langle Te_{j},e_{i}\rangle e_{i},
-\]
+ es 
+\series bold
+ortogonal
+\series default
+ a 
+\begin_inset Formula $M\subseteq H$
+\end_inset
 
+, 
+\begin_inset Formula $x\bot M$
 \end_inset
 
-con lo que 
-\begin_inset Formula $T$
+, si 
+\begin_inset Formula $\forall y\in M,x\bot y$
 \end_inset
 
- admite una representación matricial 
-\begin_inset Formula $(\langle Te_{j},e_{i}\rangle)_{i,j}\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+, y llamamos 
+\begin_inset Formula $M^{\bot}\coloneqq\{x\in H:x\bot M\}$
 \end_inset
 
 .
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula $T\in L(X,Y)$
+ Una familia 
+\begin_inset Formula $\{x_{i}\}_{i\in I}\subseteq H$
 \end_inset
 
  es 
 \series bold
-de rango finito
+ortogonal
 \series default
  si 
-\begin_inset Formula $\dim\text{Im}T<\infty$
+\begin_inset Formula $\forall i,j\in I,(i\neq j\implies x_{i}\bot x_{j})$
 \end_inset
 
-.
- Dados espacios de Hilbert 
-\begin_inset Formula $G$
+, y es 
+\series bold
+ortonormal
+\series default
+ si además 
+\begin_inset Formula $\forall i,\Vert x_{i}\Vert=1$
 \end_inset
 
- y 
-\begin_inset Formula $H$
-\end_inset
+.
+ Entonces:
+\end_layout
 
- y 
-\begin_inset Formula $T\in L(G,H)$
+\begin_layout Enumerate
+
+\series bold
+Teorema de Pitágoras:
+\series default
+ Si 
+\begin_inset Formula $x\bot y$
 \end_inset
 
 , 
-\begin_inset Formula $T$
+\begin_inset Formula $\Vert x+y\Vert^{2}=\Vert x\Vert^{2}+\Vert y\Vert^{2}$
 \end_inset
 
- es de rango finito si y sólo si viene dada por 
-\begin_inset Formula $T(x)=\sum_{i=1}^{n}\langle x,u_{i}\rangle v_{i}$
-\end_inset
+.
+\begin_inset Note Note
+status open
 
- para ciertos 
-\begin_inset Formula $u_{1},\dots,u_{n}\in G$
-\end_inset
+\begin_layout Plain Layout
+nproof
+\end_layout
 
- y 
-\begin_inset Formula $v_{1},\dots,v_{n}\in H$
 \end_inset
 
-, en cuyo caso los 
-\begin_inset Formula $(v_{i})_{i}$
-\end_inset
 
- pueden tomarse de forma que sean una base de 
-\begin_inset Formula $\text{Im}T$
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(x_{i})_{i\in I}$
 \end_inset
 
-.
+ es una familia ortogonal de elementos no nulos, es una familia linealmente
+ independiente.
 \begin_inset Note Note
 status open
 
@@ -553,102 +668,90 @@ nproof
 
 \end_layout
 
-\begin_layout Section
-Inversión de operadores
-\end_layout
-
-\begin_layout Standard
+\begin_layout Enumerate
 Si 
-\begin_inset Formula $X$
+\begin_inset Formula $M\subseteq H$
 \end_inset
 
- e 
-\begin_inset Formula $Y$
+, 
+\begin_inset Formula $M^{\bot}$
 \end_inset
 
- son 
-\begin_inset Formula $\mathbb{K}$
+ es un subespacio cerrado de 
+\begin_inset Formula $H$
 \end_inset
 
--espacios normados, 
-\begin_inset Formula $T\in{\cal L}(X,Y)$
-\end_inset
+.
+\begin_inset Note Note
+status open
 
- y 
-\begin_inset Formula $S\in{\cal L}(Y,X)$
-\end_inset
+\begin_layout Plain Layout
+nproof
+\end_layout
 
- cumplen 
-\begin_inset Formula $ST=1_{X}$
 \end_inset
 
- entonces 
-\begin_inset Formula $S$
-\end_inset
 
- es el 
+\end_layout
+
+\begin_layout Standard
+
 \series bold
-inverso por la izquierda
+Lema de Gram-Schmidt:
 \series default
- de 
-\begin_inset Formula $T$
+ Sean 
+\begin_inset Formula $H$
 \end_inset
 
- y 
-\begin_inset Formula $T$
+ prehilbertiano, 
+\begin_inset Formula $\{x_{n}\}_{n}\subseteq H$
 \end_inset
 
- es el 
-\series bold
-inverso por la derecha
-\series default
- de 
-\begin_inset Formula $S$
+ una familia contable linealmente independiente y 
+\begin_inset Formula $(u_{n})_{n}$
 \end_inset
 
-, y 
-\begin_inset Formula $T\in{\cal L}(X,Y)$
+ e 
+\begin_inset Formula $(y_{n})_{n}$
 \end_inset
 
- es 
-\series bold
-invertible
-\series default
- si existe 
-\begin_inset Formula $T^{-1}\in{\cal L}(Y,X)$
+ dadas por 
+\begin_inset Formula $u_{n}\coloneqq\frac{y_{n}}{\Vert y_{n}\Vert}$
 \end_inset
 
- inverso de 
-\begin_inset Formula $T$
+, 
+\begin_inset Formula $y_{0}\coloneqq x_{0}$
 \end_inset
 
- por la izquierda y por la derecha.
- Llamamos 
-\begin_inset Formula ${\cal L}(X)\coloneqq\text{End}_{\mathbb{K}}X={\cal L}(X,X)$
+ y para 
+\begin_inset Formula $n\geq1$
 \end_inset
 
- e 
+,
 \begin_inset Formula 
 \[
-\text{Isom}X\coloneqq\text{Isom}_{\mathbb{K}}(X)\coloneqq\{T\in{\cal L}(X)\mid T\text{ invertible}\}.
+y_{n}\coloneqq x_{n}-\sum_{j<n}\langle x_{n},u_{j}\rangle u_{j},
 \]
 
 \end_inset
 
 
-\end_layout
+\begin_inset Formula $(u_{n})_{n}$
+\end_inset
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $X$
+ es una sucesión ortonormal en 
+\begin_inset Formula $H$
 \end_inset
 
- es de dimensión finita, 
-\begin_inset Formula $T\in{\cal L}(X)$
+ y, para cada 
+\begin_inset Formula $n$
+\end_inset
+
+, 
+\begin_inset Formula $\text{span}\{u_{1},\dots,u_{n}\}=\text{span}\{x_{1},\dots,x_{n}\}$
 \end_inset
 
- tiene inverso por la izquierda si y sólo si lo tiene por la derecha, si
- y sólo si es invertible.
+.
 \begin_inset Note Note
 status open
 
@@ -658,129 +761,98 @@ nproof
 
 \end_inset
 
- Esto no es cierto en general en dimensión infinita; por ejemplo, el operador
- 
-\series bold
-desplazamiento a derecha
-\series default
-, 
-\begin_inset Formula $S_{\text{r}}\in\ell^{2}$
-\end_inset
 
- dado por 
-\begin_inset Formula $S_{\text{r}}(x_{1},\dots,x_{n},\dots)\coloneqq(0,x_{1},\dots,x_{n},\dots)$
-\end_inset
+\end_layout
 
-, tiene como inverso por la izquierda el 
-\series bold
-desplazamiento a izquierda
-\series default
-, 
-\begin_inset Formula $S_{\text{l}}\in\ell^{2}$
+\begin_layout Standard
+Si 
+\begin_inset Formula $M$
 \end_inset
 
- dado por 
-\begin_inset Formula $S_{\text{l}}(x_{1},\dots,x_{n},\dots)\coloneqq(x_{2},\dots,x_{n},\dots)$
+ es un subespacio de dimensión finita del espacio prehilbertiano 
+\begin_inset Formula $H$
 \end_inset
 
-, pero no tiene inverso por la derecha.
+:
 \end_layout
 
-\begin_layout Standard
-Sea 
-\begin_inset Formula $T\in\text{End}_{\mathbb{K}}X$
+\begin_layout Enumerate
+\begin_inset Formula $M$
 \end_inset
 
-, 
-\begin_inset Formula $\lambda\in\mathbb{K}$
-\end_inset
+ tiene una base algebraica formada por vectores ortonormales.
+\begin_inset Note Note
+status open
 
- es un 
-\series bold
-valor regular
-\series default
- de 
-\begin_inset Formula $T$
-\end_inset
+\begin_layout Plain Layout
+nproof
+\end_layout
 
- si 
-\begin_inset Formula $T-\lambda1_{X}$
 \end_inset
 
- es invertible, un 
-\series bold
-valor espectral
-\series default
- en otro caso, y un 
-\series bold
-valor propio
-\series default
- si 
-\begin_inset Formula $\ker(T-\lambda1_{X})\neq0$
-\end_inset
 
-, en cuyo caso llamamos 
-\series bold
-subespacio propio
-\series default
- de 
-\begin_inset Formula $T$
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $M$
 \end_inset
 
- correspondiente al valor propio 
-\begin_inset Formula $\lambda$
+ es equivalente a 
+\begin_inset Formula $\mathbb{K}^{\dim M}$
 \end_inset
 
- a 
-\begin_inset Formula $\ker(T-\lambda1_{X})$
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
- y 
+
+\end_layout
+
+\begin_layout Standard
+Un 
 \series bold
-valores propios
+espacio de Hilbert
 \series default
- de 
-\begin_inset Formula $T$
+ es un espacio prehilbertiano completo.
+ Dado un espacio de medida 
+\begin_inset Formula $(\Omega,\Sigma,\mu)$
 \end_inset
 
- correspondientes al valor propio 
-\begin_inset Formula $\lambda$
+, 
+\begin_inset Formula $L^{2}(\Omega,\Sigma,\mu)$
 \end_inset
 
- a los elementos no nulos de este subespacio.
- Llamamos 
-\series bold
-resolvente
-\series default
- de 
-\begin_inset Formula $T$
+ es un espacio de Hilbert con
+\begin_inset Formula 
+\[
+\langle f,g\rangle\coloneqq\int_{\Omega}f\overline{g}\dif\mu,
+\]
+
 \end_inset
 
- al conjunto de sus valores regulares, 
-\series bold
-espectro
-\series default
- de 
-\begin_inset Formula $T$
+y en particular lo son 
+\begin_inset Formula $\ell^{2}$
 \end_inset
 
-, 
-\begin_inset Formula $\sigma(T)$
+ con 
+\begin_inset Formula $\langle x,y\rangle\coloneqq\sum_{n}x_{n}\overline{y_{n}}$
 \end_inset
 
-, al conjunto de sus valores espectrales y 
-\series bold
-espectro puntual
-\series default
- de 
-\begin_inset Formula $T$
+ y 
+\begin_inset Formula $\ell_{n}^{2}$
 \end_inset
 
-, 
-\begin_inset Formula $\sigma_{\text{p}}(T)\subseteq\sigma(T)$
+ con 
+\begin_inset Formula $\langle x,y\rangle\coloneqq\sum_{i}x_{i}\overline{y_{i}}$
 \end_inset
 
-, al conjunto de sus valores propios.
+.
 \begin_inset Note Note
 status open
 
@@ -794,12 +866,15 @@ nproof
 \end_layout
 
 \begin_layout Standard
-Si 
-\begin_inset Formula $X$
+Son espacios prehilbertianos no completos:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $c_{00}$
 \end_inset
 
- es de dimensión finita, 
-\begin_inset Formula $\sigma_{\text{p}}(T)=\sigma(T)$
+ con el producto escalar de 
+\begin_inset Formula $\ell^{2}$
 \end_inset
 
 .
@@ -812,12 +887,23 @@ nproof
 
 \end_inset
 
- Sin embargo, 
-\begin_inset Formula $0\in\sigma(S_{\text{r}})$
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $C([a,b])$
+\end_inset
+
+ con el producto escalar de 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ con la medida de Lebesgue, y entonces 
+\begin_inset Formula $C([a,b])$
 \end_inset
 
- pero 
-\begin_inset Formula $\sigma_{\text{p}}(S_{\text{r}})=\emptyset$
+ es denso en 
+\begin_inset Formula $L^{2}([a,b])$
 \end_inset
 
 .
@@ -833,33 +919,90 @@ nproof
 
 \end_layout
 
+\begin_layout Section
+Mejor aproximación
+\end_layout
+
 \begin_layout Standard
-Como 
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio vectorial, 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es 
 \series bold
-teorema
+convexo
 \series default
-, si 
+ si 
+\begin_inset Formula $\forall\lambda\in[0,1]$
+\end_inset
+
+, 
+\begin_inset Formula $\lambda A+(1-\lambda)A\subseteq A$
+\end_inset
+
+.
+ Si 
 \begin_inset Formula $X$
 \end_inset
 
- es un espacio de Banach y 
-\begin_inset Formula $T\in{\cal L}(X)$
+ es normado, 
+\begin_inset Formula $S\subseteq X$
 \end_inset
 
- cumple 
-\begin_inset Formula $\Vert T\Vert<1$
+ no vacío y 
+\begin_inset Formula $x\in X$
 \end_inset
 
-, 
-\begin_inset Formula $1_{X}-T$
+, un 
+\begin_inset Formula $y\in S$
+\end_inset
+
+ es un 
+\series bold
+vector de mejor aproximación
+\series default
+ de 
+\begin_inset Formula $x$
 \end_inset
 
- es invertible con inverso 
-\begin_inset Formula $\sum_{n\in\mathbb{N}}T^{n}$
+ a 
+\begin_inset Formula $S$
 \end_inset
 
- y 
-\begin_inset Formula $\Vert(1_{X}-T)^{-1}\Vert\leq\frac{1}{1-\Vert T\Vert}$
+ si 
+\begin_inset Formula $\Vert x-y\Vert=\min_{z\in S}\Vert x-z\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de mejor aproximación:
+\series default
+ Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio prehilbertiano y 
+\begin_inset Formula $C\subseteq H$
+\end_inset
+
+ es no vacío, convexo y completo, para cada 
+\begin_inset Formula $x\in H$
+\end_inset
+
+ existe una mejor aproximación de 
+\begin_inset Formula $x$
+\end_inset
+
+ a 
+\begin_inset Formula $C$
 \end_inset
 
 .
@@ -867,42 +1010,77 @@ teorema
 \series bold
 Demostración:
 \series default
+ Podemos suponer por traslación que 
+\begin_inset Formula $x=0$
+\end_inset
+
+, y llamamos 
+\begin_inset Formula $\alpha\coloneqq\inf_{z\in C}\Vert z\Vert$
+\end_inset
+
+.
+ Para la existencia tomamos una sucesión 
+\begin_inset Formula $\{y_{n}\}_{n}\subseteq C$
+\end_inset
+
+ con 
+\begin_inset Formula $\lim_{n}\Vert y_{n}\Vert=\alpha$
+\end_inset
+
+ y probamos que es de Cauchy, pues entonces por completitud existe 
+\begin_inset Formula $y\coloneqq\lim_{n}y_{n}\in C$
+\end_inset
+
+ y por continuidad de la norma es 
+\begin_inset Formula $\Vert y\Vert=\alpha$
+\end_inset
+
+.
  Para 
-\begin_inset Formula $n\in\mathbb{N}$
+\begin_inset Formula $\varepsilon>0$
 \end_inset
 
-, 
-\begin_inset Formula $\sum_{k=0}^{n}\Vert T^{k}\Vert\leq\sum_{k=0}^{n}\Vert T\Vert^{k}\leq\sum_{k\in\mathbb{N}}\Vert T\Vert^{n}=\frac{1}{1-\Vert T\Vert}$
+ existe 
+\begin_inset Formula $n_{0}$
 \end_inset
 
-, con lo que 
-\begin_inset Formula $\sum_{n}\Vert T^{n}\Vert$
+ tal que si 
+\begin_inset Formula $n\geq n_{0}$
 \end_inset
 
- converge y, por ser 
-\begin_inset Formula $X$
+ es 
+\begin_inset Formula $\Vert y_{n}\Vert^{2}<\alpha^{2}+\varepsilon$
 \end_inset
 
- de Banach, 
-\begin_inset Formula $S\coloneqq\sum_{n}T^{n}$
+, y por la ley del paralelogramo es
+\begin_inset Formula 
+\[
+\left\Vert \frac{y_{n}-y_{m}}{2}\right\Vert ^{2}=\frac{1}{2}(\Vert y_{n}\Vert^{2}+\Vert y_{m}\Vert^{2})-\left\Vert \frac{y_{n}+y_{m}}{2}\right\Vert ^{2}\leq\frac{1}{2}(\alpha^{2}+\varepsilon+\alpha^{2}+\varepsilon)-\alpha^{2}=\varepsilon,
+\]
+
 \end_inset
 
- también, pero 
-\begin_inset Formula $S(1_{X}-T)=S-ST=T^{0}=1_{X}$
+pues por convexidad 
+\begin_inset Formula $\frac{y_{n}+y_{m}}{2}\in S$
 \end_inset
 
- y análogamente 
-\begin_inset Formula $(1_{X}-T)S=1_{X}$
+ y por tanto su norma es mayor o igual a 
+\begin_inset Formula $\alpha$
 \end_inset
 
-, luego 
-\begin_inset Formula $S=(1_{X}-T)^{-1}$
+.
+ Para la unicidad, si 
+\begin_inset Formula $y,z\in C$
 \end_inset
 
-, y finalmente
+ cumplen 
+\begin_inset Formula $\Vert y\Vert=\Vert z\Vert=\alpha$
+\end_inset
+
+, por un argumento como el anterior,
 \begin_inset Formula 
 \[
-\Vert(1_{X}-T)^{-1}\Vert=\left\Vert \sum_{n}T^{n}\right\Vert \leq\sum_{n}\Vert T\Vert^{n}=\frac{1}{1-\Vert T\Vert}.
+\left\Vert \frac{y-z}{2}\right\Vert ^{2}=\frac{1}{2}(\Vert y\Vert^{2}+\Vert z\Vert^{2})-\left\Vert \frac{y+z}{2}\right\Vert ^{2}\leq\frac{1}{2}(\alpha^{2}+\alpha^{2})-\alpha^{2}=0.
 \]
 
 \end_inset
@@ -911,674 +1089,880 @@ Demostración:
 \end_layout
 
 \begin_layout Standard
-
+Como 
 \series bold
-Teorema de von Neumann:
+teorema
 \series default
- Sean 
-\begin_inset Formula $X$
+, si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es un subespacio de un espacio prehilbertiano 
+\begin_inset Formula $H$
 \end_inset
 
- es un espacio de Banach, 
-\begin_inset Formula $T\in{\cal L}(X)$
+ y 
+\begin_inset Formula $x\in H$
 \end_inset
 
- invertible y 
-\begin_inset Formula $S\in{\cal L}(X)$
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $y\in Y$
 \end_inset
 
- tal que 
-\begin_inset Formula $\Vert T-S\Vert<\frac{1}{\Vert T^{-1}\Vert}$
+ es de mejor aproximación de 
+\begin_inset Formula $x$
 \end_inset
 
-, entonces 
-\begin_inset Formula $S$
+ a 
+\begin_inset Formula $Y$
 \end_inset
 
- es invertible con
-\begin_inset Formula 
-\begin{align*}
-S^{-1} & =\sum_{n\in\mathbb{N}}(T^{-1}(T-S))^{n}T^{-1}, & \left\Vert T^{-1}-S^{-1}\right\Vert  & \leq\frac{\Vert T^{-1}\Vert^{2}\Vert T-S\Vert}{1-\Vert T^{-1}\Vert\Vert T-S\Vert}.
-\end{align*}
+ si y sólo si 
+\begin_inset Formula $x-y\bot Y$
+\end_inset
+
+.
+\end_layout
 
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
 \end_inset
 
 
-\series bold
-Demostración:
-\series default
- 
-\begin_inset Formula $\Vert T^{-1}(T-S)\Vert=\Vert T-S\Vert\Vert T^{-1}\Vert<1$
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $z\in Y$
 \end_inset
 
-, luego por el teorema anterior 
-\begin_inset Formula $1_{X}-T^{-1}(T-S)=T^{-1}S$
+ y 
+\begin_inset Formula $a\in\mathbb{K}$
+\end_inset
+
+, como 
+\begin_inset Formula $y-az\in Y$
 \end_inset
 
- es invertible con
+,
 \begin_inset Formula 
 \[
-(T^{-1}S)^{-1}=\sum_{n}(T^{-1}(T-S))^{n},
+\Vert x-y\Vert^{2}\leq\Vert x-y+az\Vert^{2}=\Vert x-y\Vert^{2}+2\text{Re}(a\langle z,x-y\rangle)+|a|^{2}\Vert z\Vert^{2},
 \]
 
 \end_inset
 
 luego 
-\begin_inset Formula $S=T(T^{-1}S)$
+\begin_inset Formula $0\leq2\text{Re}(a\langle z,x-y\rangle)+|a|^{2}\Vert z\Vert^{2}$
 \end_inset
 
- es invertible con inversa 
-\begin_inset Formula $(T^{-1}S)^{-1}T^{-1}$
+ y, haciendo 
+\begin_inset Formula $a=t\langle x-y,z\rangle$
 \end_inset
 
- y
-\begin_inset Formula 
-\begin{align*}
-\Vert T^{-1}-S^{-1}\Vert & =\Vert T^{-1}-(T^{-1}S)^{-1}T^{-1}\Vert=\Vert(1_{X}-(T^{-1}S)^{-1})T^{-1}\Vert\leq\\
- & \leq\left\Vert \left(1_{X}-\sum_{n}(T^{-1}(T-S))^{n}\right)T^{-1}\right\Vert =\left\Vert \sum_{n\geq1}(T^{-1}(T-S))^{n}T^{-1}\right\Vert \leq\\
- & \leq\sum_{n\geq1}\Vert(T^{-1}(T-S))^{n}\Vert\Vert T^{-1}\Vert\leq\frac{\Vert T^{-1}\Vert^{2}\Vert T-S\Vert}{1-\Vert T^{-1}\Vert\Vert T-S\Vert}.
-\end{align*}
+ con 
+\begin_inset Formula $t\in\mathbb{R}$
+\end_inset
 
+, 
+\begin_inset Formula $0\leq2t|\langle x-y,z\rangle|^{2}+t^{2}|\langle x-y,z\rangle|^{2}\Vert z\Vert^{2}$
 \end_inset
 
+.
+ Si hubiera 
+\begin_inset Formula $z\in Y$
+\end_inset
 
-\end_layout
+ con 
+\begin_inset Formula $\langle x-y,z\rangle\neq0$
+\end_inset
 
-\begin_layout Standard
-Así, si 
-\begin_inset Formula $X$
+, 
+\begin_inset Formula $0\leq2t+t^{2}\Vert z\Vert^{2}$
 \end_inset
 
- es un espacio de Banach, 
-\begin_inset Formula $\text{Isom}X$
+ para todo 
+\begin_inset Formula $t\in\mathbb{R}$
 \end_inset
 
- es un abierto de 
-\begin_inset Formula ${\cal L}(X)$
+, pero si 
+\begin_inset Formula $\Vert z\Vert^{2}=0$
 \end_inset
 
- y 
-\begin_inset Formula $\cdot^{-1}:\text{Isom}X\to\text{Isom}X$
+, esto es negativo cuando 
+\begin_inset Formula $t<0$
+\end_inset
+
+, y si 
+\begin_inset Formula $\Vert z\Vert^{2}>0$
+\end_inset
+
+, es negativo al menos cuando 
+\begin_inset Formula $t=-\frac{1}{\Vert z\Vert^{2}}\#$
+\end_inset
+
+, luego 
+\begin_inset Formula $x-y\bot z$
 \end_inset
 
- es continua con la norma de 
-\begin_inset Formula ${\cal L}(X)$
+ y 
+\begin_inset Formula $x-y\bot Y$
 \end_inset
 
 .
 \end_layout
 
-\begin_layout Standard
-\begin_inset ERT
+\begin_layout Enumerate
+\begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
-
-
-\backslash
-begin{reminder}{FVC}
-\end_layout
-
+\begin_inset Formula $\impliedby]$
 \end_inset
 
 
 \end_layout
 
-\begin_layout Standard
+\end_inset
 
-\series bold
-Teorema de Liouville:
-\series default
- Toda función [...][compleja holomorfa y] acotada es constante.
-\end_layout
+Para 
+\begin_inset Formula $z\in Y$
+\end_inset
 
-\begin_layout Standard
-\begin_inset ERT
-status open
+, por el teorema de Pitágoras,
+\begin_inset Formula 
+\[
+\Vert x-z\Vert^{2}=\Vert x-y+y-z\Vert^{2}=\Vert x-y\Vert^{2}+\Vert y-z\Vert^{2}\geq\Vert x-y\Vert^{2}.
+\]
 
-\begin_layout Plain Layout
+\end_inset
 
 
-\backslash
-end{reminder}
 \end_layout
 
+\end_deeper
+\begin_layout Enumerate
+Si existe una mejor aproximación de 
+\begin_inset Formula $x$
 \end_inset
 
+ a 
+\begin_inset Formula $Y$
+\end_inset
 
+, es única.
 \end_layout
 
+\begin_deeper
 \begin_layout Standard
+Sean 
+\begin_inset Formula $y,z\in Y$
+\end_inset
 
-\series bold
-Teorema de Gelfand:
-\series default
- Si 
-\begin_inset Formula $_{\mathbb{C}}X$
+ de mejor aproximación, como 
+\begin_inset Formula $x-y,x-z\in Y^{\bot}$
 \end_inset
 
- es de Banach y 
-\begin_inset Formula $T\in{\cal L}(X)$
+, su diferencia 
+\begin_inset Formula $y-z\in Y^{\bot}\cap Y$
 \end_inset
 
-, 
-\begin_inset Formula $\sigma(T)$
+, luego 
+\begin_inset Formula $\langle y-z,y-z\rangle=0$
 \end_inset
 
- es compacto no vacío contenido en 
-\begin_inset Formula $B(0,\Vert T\Vert)$
+ e 
+\begin_inset Formula $y=z$
 \end_inset
 
 .
- 
-\series bold
-Demostración:
-\series default
- Si 
-\begin_inset Formula $\lambda\in\mathbb{C}\setminus B[0,\Vert T\Vert]$
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $Y$
 \end_inset
 
-, 
-\begin_inset Formula $\frac{\Vert T\Vert}{|\lambda|}<1$
+ es completo, hay vector de mejor aproximación.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Por el teorema anterior (los subespacios son convexos).
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Determinante de Gram
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $H$
 \end_inset
 
-, luego 
-\begin_inset Formula $\lambda1_{X}-T=\lambda(1_{X}-\frac{T}{\lambda})$
+ prehilbertiano y 
+\begin_inset Formula $M\leq H$
 \end_inset
 
- es invertible y 
-\begin_inset Formula $\lambda\notin\sigma(T)$
+ de dimensión finita con base ortonormal 
+\begin_inset Formula $(e_{i})_{i}$
 \end_inset
 
 .
- La función 
-\begin_inset Formula $\psi:\mathbb{C}\to{\cal L}(X)$
-\end_inset
+\end_layout
 
- dada por 
-\begin_inset Formula $\psi(\lambda)\coloneqq\lambda1_{X}-T$
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $x\in H$
 \end_inset
 
- es continua y por tanto 
-\begin_inset Formula $\mathbb{C}\setminus\sigma(T)=\psi^{-1}(\text{Isom}X)$
+ existe un único vector de aproximación de 
+\begin_inset Formula $x$
 \end_inset
 
- es abierto, con lo que 
-\begin_inset Formula $\sigma(T)$
+ a 
+\begin_inset Formula $M$
 \end_inset
 
- es cerrado acotado y por tanto compacto.
- Si fuera vacío, podemos definir 
-\begin_inset Formula $\phi:\mathbb{C}\to\text{Isom}X$
-\end_inset
+ dado por
+\begin_inset Formula 
+\[
+\sum_{i}\langle x,e_{i}\rangle e_{i}.
+\]
 
- como 
-\begin_inset Formula $\phi(\lambda)\coloneqq(\lambda1_{X}-T)^{-1}$
 \end_inset
 
-, que es continua, pero para 
-\begin_inset Formula $\lambda,h\in\mathbb{C}$
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $d(x,M)^{2}=\Vert x\Vert^{2}-\sum_{i}|\langle x,e_{i}\rangle|^{2}$
 \end_inset
 
-,
-\begin_inset Formula 
-\begin{multline*}
-\frac{\phi(\lambda+h)-\phi(\lambda)}{h}=\frac{((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1}((\lambda1_{X}-T)-((\lambda+h)1_{X}-T))}{h}=\\
-=-((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1},
-\end{multline*}
+.
+\end_layout
 
+\begin_layout Standard
+Llamamos 
+\series bold
+determinante de Gram
+\series default
+ de 
+\begin_inset Formula $(x_{i})_{i=1}^{n}$
 \end_inset
 
-de donde
+ a
 \begin_inset Formula 
 \[
-\dot{\phi}(\lambda)=\lim_{h\to0}\frac{\phi(\lambda+h)-\phi(\lambda)}{h}=\lim_{h\to0}(-((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1})=-((\lambda1_{X}-T)^{-1})^{2},
+G(x_{1},\dots,G_{n})\coloneqq\det(\langle x_{j},x_{i}\rangle)_{1\leq i\leq n}^{1\leq j\leq n}.
 \]
 
 \end_inset
 
-con lo que 
-\begin_inset Formula $\phi$
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $H$
+\end_inset
+
+ es prehilbertiano, 
+\begin_inset Formula $M\leq H$
+\end_inset
+
+ de dimensión finita con base 
+\begin_inset Formula $(b_{i})_{i}$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in H$
+\end_inset
+
+, el vector de mejor aproximación de 
+\begin_inset Formula $x$
 \end_inset
 
- es holomorfa y 
-\begin_inset Formula $\dot{\phi}\neq0$
+ a 
+\begin_inset Formula $M$
 \end_inset
 
-, pero
+ es
 \begin_inset Formula 
 \[
-\Vert\phi(\lambda)\Vert=\Vert(\lambda1_{X}-T)^{-1}\Vert=\frac{1}{|\lambda|}\left\Vert \left(1_{X}-\frac{T}{\lambda}\right)^{-1}\right\Vert =\frac{1}{|\lambda|}\left\Vert \sum_{n\in\mathbb{N}}\frac{T^{n}}{\lambda^{n}}\right\Vert \leq\frac{1}{|\lambda|}\frac{1}{1-\frac{\Vert T\Vert}{|\lambda|}}=\frac{1}{|\lambda|-\Vert T\Vert},
+\frac{-1}{G(b_{1},\dots,b_{n})}\begin{vmatrix}\langle x_{1},x_{1}\rangle & \langle x_{2},x_{1}\rangle & \cdots & \langle x_{n},x_{1}\rangle & \langle x,x_{1}\rangle\\
+\langle x_{1},x_{2}\rangle & \langle x_{2},x_{2}\rangle & \cdots & \langle x_{n},x_{2}\rangle & \langle x,x_{2}\rangle\\
+\vdots & \vdots & \ddots & \vdots & \vdots\\
+\langle x_{1},x_{n}\rangle & \langle x_{2},x_{n}\rangle & \cdots & \langle x_{n},x_{n}\rangle & \langle x,x_{n}\rangle\\
+x_{1} & x_{2} & \cdots & x_{n} & 0
+\end{vmatrix},
 \]
 
 \end_inset
 
-con lo que 
-\begin_inset Formula $\lim_{|\lambda|\to\infty}\Vert\phi(\lambda)\Vert=\infty$
-\end_inset
+y
+\begin_inset Formula 
+\[
+d(x,M)=\sqrt{\frac{G(x_{1},\dots,x_{n},x)}{G(x_{1},\dots,x_{n})}}.
+\]
 
- y por tanto, como 
-\begin_inset Formula $\phi$
 \end_inset
 
- es continua, es acotada y, por el teorema de Liouville
-\begin_inset Foot
+
+\begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-Que todavía no hemos visto que se de para espacios vectoriales infinitos
- pero suponemos que se cumple.
+nproof
 \end_layout
 
 \end_inset
 
-, 
-\begin_inset Formula $\phi$
+
+\end_layout
+
+\begin_layout Standard
+Algunas aplicaciones:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Resolución de sistemas sobre-dimensionados por mínimos cuadrados.
+
+\series default
+ Tenemos un fenómeno experimental que se puede modelar como una función
+ lineal 
+\begin_inset Formula $y(x)=a_{1}x_{1}+\dots+a_{n}x_{n}$
 \end_inset
 
- es constante y 
-\begin_inset Formula $\dot{\phi}=0\#$
+, pero no conocemos los 
+\begin_inset Formula $a_{i}$
 \end_inset
 
 .
-\end_layout
+ Hacemos 
+\begin_inset Formula $m$
+\end_inset
 
-\begin_layout Standard
-Dados 
-\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+ experimentos fijando un 
+\begin_inset Formula $x_{i}$
 \end_inset
 
- con 
-\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<1$
+ en cada uno y midiendo 
+\begin_inset Formula $y_{i}\coloneqq y(x_{i})$
 \end_inset
 
- e 
-\begin_inset Formula $y\in\ell^{2}$
+ para plantear un sistema de 
+\begin_inset Formula $m$
 \end_inset
 
-, el sistema
-\begin_inset Formula 
-\begin{align*}
-x_{k}-\sum_{j\in\mathbb{N}}a_{kj}x_{j} & =y_{k}, & k & \in\mathbb{N},
-\end{align*}
+ ecuaciones.
+ Solo hacen falta 
+\begin_inset Formula $n$
+\end_inset
 
+ experimentos cuidando que los 
+\begin_inset Formula $x_{i}$
 \end_inset
 
-tiene solución única 
-\begin_inset Formula $z\in\ell^{2}$
+ sean linealmente independientes, pero en general conviene hacer más, 
+\begin_inset Formula $m>n$
 \end_inset
 
-, y para 
-\begin_inset Formula $n\in\mathbb{N}$
+.
+ Como las mediciones son aproximadas, el sistema puede ser incompatible,
+ por lo que se eligen los 
+\begin_inset Formula $a_{i}\in\mathbb{R}$
 \end_inset
 
-, el sistema truncado
+ de forma que se minimice
 \begin_inset Formula 
-\begin{align*}
-x_{k}-\sum_{j\in\mathbb{N}_{n}}a_{kj}x_{j} & =y_{k}, & k & \in\mathbb{N}_{n}
-\end{align*}
+\[
+\sum_{i\in\mathbb{N}_{m}}\left(y_{i}-\sum_{j\in\mathbb{N}_{n}}a_{j}x_{ij}\right)^{2}=\left\Vert y-\sum_{j\in\mathbb{N}_{n}}a_{j}X_{j}\right\Vert ^{2},
+\]
 
 \end_inset
 
-tiene una única solución 
-\begin_inset Formula $z_{n}\in\mathbb{K}^{n}$
+donde 
+\begin_inset Formula $X_{j}\coloneqq(x_{1j},\dots,x_{mj})$
 \end_inset
 
- de modo que, si 
-\begin_inset Formula $J_{n}:\mathbb{K}^{n}\to\ell^{2}$
+.
+ Si 
+\begin_inset Formula $X_{1},\dots,X_{n}$
 \end_inset
 
- es la inclusión canónica de 
-\begin_inset Formula $\mathbb{K}^{n}$
+ son linealmente independientes, sea 
+\begin_inset Formula $M\coloneqq\text{span}\{X_{1},\dots,X_{n}\}<\mathbb{R}^{m}$
 \end_inset
 
- en las 
-\begin_inset Formula $n$
+, buscamos el vector 
+\begin_inset Formula $Z\in M$
 \end_inset
 
- primeras coordenadas, 
-\begin_inset Formula $\lim_{n}J_{n}(z_{n})=z$
+ de mejor aproximación de 
+\begin_inset Formula $y$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
+ en 
+\begin_inset Formula $M$
+\end_inset
 
+ que, expresado respecto de la base 
+\begin_inset Formula $(X_{1},\dots,X_{n})$
 \end_inset
 
+, nos dará el vector 
+\begin_inset Formula $(a_{1},\dots,a_{n})$
+\end_inset
 
+ buscado.
 \end_layout
 
-\begin_layout Standard
-Sean 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\begin_layout Enumerate
+
+\series bold
+Ajustes polinómicos por mínimos cuadrados.
+
+\series default
+ Queremos modelar un fenómeno experimental como una función polinómica 
+\begin_inset Formula $f:[a,b]\to\mathbb{R}$
+\end_inset
+
+, y tenemos 
+\begin_inset Formula $k$
+\end_inset
+
+ observaciones de la forma 
+\begin_inset Formula $f(t_{i})=y_{i}$
 \end_inset
 
  con 
-\begin_inset Formula $\Vert k\Vert_{2}<1$
+\begin_inset Formula $t_{1}<\dots<t_{k}$
 \end_inset
 
- y 
-\begin_inset Formula $g\in L^{2}([a,b])$
+.
+ Existe un polinomio de grado máximo 
+\begin_inset Formula $k-1$
 \end_inset
 
-, la ecuación
-\begin_inset Formula 
-\begin{align*}
-f(t)-\int_{a}^{b}k(t,s)f(s)\dif s & =g(t), & t & \in[a,b],
-\end{align*}
+ que cumple esto, pero muchas veces 
+\begin_inset Formula $k$
+\end_inset
+
+ es muy grande y esto complica los cálculos y puede llevar al 
+\emph on
+\lang english
+overfitting
+\emph default
+\lang spanish
+ o fenómeno de Runge.
+ Entonces buscamos un polinomio 
+\begin_inset Formula $f$
+\end_inset
 
+ de grado máximo 
+\begin_inset Formula $n$
+\end_inset
+
+ bastante menor que 
+\begin_inset Formula $k-1$
 \end_inset
 
-tiene solución única que es de la forma
+ que minimice
 \begin_inset Formula 
 \[
-g(t)+\int_{a}^{b}\tilde{k}(t,s)g(s)\dif s
+\sum_{i\in\mathbb{N}_{k}}|y_{i}-f(t_{i})|^{2}=\left\Vert y-\sum_{j=0}^{n}f_{j}t^{j}\right\Vert ^{2},
 \]
 
 \end_inset
 
-para cierto 
-\begin_inset Formula $\tilde{k}\in L^{2}([a,b]\times[a,b])$
+donde 
+\begin_inset Formula $t^{j}\coloneqq(t_{1}^{j},\dots,t_{k}^{j})$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
+ Para ello, como para 
+\begin_inset Formula $k\geq2$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ los 
+\begin_inset Formula $t^{j}$
+\end_inset
+
+ son linealmente independientes, consideramos 
+\begin_inset Formula $M\coloneqq\text{span}\{1,t,t^{2},\dots,t^{n}\}<\mathbb{R}^{n+1}$
+\end_inset
+
+ y buscamos la mejor aproximación de 
+\begin_inset Formula $y$
+\end_inset
 
+ a 
+\begin_inset Formula $M$
 \end_inset
 
+.
+\end_layout
 
+\begin_layout Section
+Teorema de la proyección
 \end_layout
 
 \begin_layout Standard
-Si 
-\begin_inset Formula $K$
-\end_inset
 
- es el operador integral con núcleo 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\series bold
+Teorema de la proyección:
+\series default
+ Si 
+\begin_inset Formula $H$
 \end_inset
 
-, 
-\begin_inset Formula $\Vert k\Vert_{2}<1$
+ es un espacio de Hilbert con un subespacio cerrado 
+\begin_inset Formula $M$
 \end_inset
 
  y 
-\begin_inset Formula 
-\[
-\forall t\in[a,b],\int_{a}^{b}|k(t,s)|^{2}\dif s\leq C,
-\]
-
+\begin_inset Formula $P_{M}:H\to M$
 \end_inset
 
-para 
-\begin_inset Formula $g\in L^{2}([a,b])$
+ la 
+\series bold
+proyección ortogonal
+\series default
+ de 
+\begin_inset Formula $H$
 \end_inset
 
-, la serie 
-\begin_inset Formula $\sum_{n}K^{n}g$
+ sobre 
+\begin_inset Formula $M$
 \end_inset
 
- converge en 
-\begin_inset Formula $L^{2}([a,b])$
+ que asigna a cada 
+\begin_inset Formula $x\in H$
 \end_inset
 
- y converge absoluta y uniformemente en 
-\begin_inset Formula $[a,b]$
+ la mejor aproximación de 
+\begin_inset Formula $x$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+ a 
+\begin_inset Formula $M$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
+:
 \end_layout
 
+\begin_layout Enumerate
+\begin_inset Formula $H$
 \end_inset
 
-
-\end_layout
-
-\begin_layout Standard
-Con todo esto, para 
-\begin_inset Formula $g\in L^{2}([0,1])$
+ es suma directa topológica de 
+\begin_inset Formula $M$
 \end_inset
 
  y 
-\begin_inset Formula $\lambda\in\mathbb{R}\setminus\{1\}$
+\begin_inset Formula $M^{\bot}$
 \end_inset
 
-, la ecuación integral
-\begin_inset Formula 
-\[
-f(t)-\lambda\int_{0}^{1}\text{e}^{t-s}f(s)\dif s=g(t)
-\]
+, 
+\begin_inset Formula $P_{M}$
+\end_inset
 
+ es la proyección canónica y, si 
+\begin_inset Formula $P_{M^{\bot}}:H\to M^{\bot}$
 \end_inset
 
-tiene solución única
-\begin_inset Formula 
-\[
-f(t)=g(t)+\frac{\lambda}{1-\lambda}\int_{0}^{1}\text{e}^{t-s}g(s)\dif s.
-\]
+ es la otra proyección canónica, si 
+\begin_inset Formula $M\neq0$
+\end_inset
 
+, 
+\begin_inset Formula $\Vert P_{M}\Vert=1$
 \end_inset
 
+, y si 
+\begin_inset Formula $M^{\bot}\neq0$
+\end_inset
 
-\end_layout
+, 
+\begin_inset Formula $\Vert P_{M^{\bot}}\Vert=1$
+\end_inset
 
-\begin_layout Section
-Operador adjunto
+.
 \end_layout
 
+\begin_deeper
 \begin_layout Standard
-Si 
-\begin_inset Formula $G$
+Por la definición de producto escalar, 
+\begin_inset Formula $M^{\bot}\leq H$
 \end_inset
 
- y 
-\begin_inset Formula $H$
+.
+ Claramente 
+\begin_inset Formula $M\cap M^{\bot}=0$
 \end_inset
 
- son espacios de Hilbert y 
-\begin_inset Formula $T\in L(G,H)$
+, y para 
+\begin_inset Formula $x\in M$
 \end_inset
 
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula 
-\[
-\Vert T\Vert=\sup_{x,y\in\overline{B_{G}}}|\langle Tx,y\rangle|=\sup_{x,y\in B_{G}}|\langle Tx,y\rangle|.
-\]
-
+, como 
+\begin_inset Formula $y\coloneqq P_{M}(x)$
 \end_inset
 
+ cumple 
+\begin_inset Formula $x-y\bot M$
+\end_inset
 
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
+, 
+\begin_inset Formula $x=y+z$
+\end_inset
 
+ con 
+\begin_inset Formula $y\in M$
 \end_inset
 
+ y 
+\begin_inset Formula $z\coloneqq x-y\in M^{\bot}$
+\end_inset
 
-\end_layout
+, luego 
+\begin_inset Formula $M+M^{\bot}=H$
+\end_inset
 
-\begin_layout Enumerate
-Existe un único 
-\begin_inset Formula $T^{*}\in L(H,G)$
+ y 
+\begin_inset Formula $H$
 \end_inset
 
- tal que 
-\begin_inset Formula $\forall x\in G,\forall y\in H,\langle Tx,y\rangle\equiv\langle x,T^{*}y\rangle$
+ es suma directa algebraica de 
+\begin_inset Formula $M$
 \end_inset
 
-, el 
-\series bold
-adjunto
-\series default
- de 
-\begin_inset Formula $T$
+ y 
+\begin_inset Formula $M^{\bot}$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
+ 
+\begin_inset Formula $P_{M}$
+\end_inset
 
+ es la proyección canónica porque, si 
+\begin_inset Formula $y\in M$
 \end_inset
 
+ y 
+\begin_inset Formula $z\in M^{\bot}$
+\end_inset
 
-\end_layout
+, 
+\begin_inset Formula $(y+z)-y=z\bot M$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $\Vert T\Vert=\Vert T^{*}\Vert$
+, y por unicidad de la mejor aproximación, 
+\begin_inset Formula $P_{M}(y+z)=y$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
+ 
+\begin_inset Formula $P_{M}$
+\end_inset
 
+ y 
+\begin_inset Formula $P_{M^{\bot}}$
 \end_inset
 
+ son lineales por ser proyecciones canónicas, y para 
+\begin_inset Formula $x=y+z\in S_{H}$
+\end_inset
 
-\end_layout
+ con 
+\begin_inset Formula $y\in M$
+\end_inset
 
-\begin_layout Standard
-Sean 
-\begin_inset Formula $G$
+ y 
+\begin_inset Formula $z\in M^{\bot}$
 \end_inset
 
 , 
-\begin_inset Formula $H$
+\begin_inset Formula $\Vert x\Vert^{2}=\Vert y\Vert^{2}+\Vert z\Vert^{2}=\Vert P_{M}(x)\Vert^{2}+\Vert P_{M^{\bot}}(x)\Vert^{2}$
 \end_inset
 
  y 
-\begin_inset Formula $J$
+\begin_inset Formula $\Vert P_{M}(x)\Vert,\Vert P_{M^{\bot}}(x)\Vert\leq\Vert x\Vert=1$
 \end_inset
 
- 
-\begin_inset Formula $\mathbb{K}$
+, lo que prueba la continuidad y por tanto que 
+\begin_inset Formula $M$
 \end_inset
 
--espacios de Hilbert, 
-\begin_inset Formula $A,B\in L(G,H)$
+ es topológica.
+ Además, si 
+\begin_inset Formula $M\neq0$
 \end_inset
 
-, 
-\begin_inset Formula $C\in L(H,J)$
+, existe 
+\begin_inset Formula $y\in S_{M}$
 \end_inset
 
  y 
-\begin_inset Formula $\alpha\in\mathbb{K}$
+\begin_inset Formula $\Vert P_{M}(y)\Vert=\Vert y\Vert=1$
 \end_inset
 
-:
-\end_layout
+, luego 
+\begin_inset Formula $\Vert P_{M}\Vert=1$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $(A+B)^{*}=A^{*}+B^{*}$
+, y análogamente para 
+\begin_inset Formula $M^{\bot}$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
 \end_layout
 
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $P_{M}(H)=M$
 \end_inset
 
+, 
+\begin_inset Formula $\ker P_{M}=M^{\bot}$
+\end_inset
+
+ y 
+\begin_inset Formula $P_{M^{\bot}}=1_{H}-P_{M}$
+\end_inset
 
+.
 \end_layout
 
 \begin_layout Enumerate
-\begin_inset Formula $(\alpha A)^{*}=\overline{\alpha}A^{*}$
+Para 
+\begin_inset Formula $x,y\in H$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+, 
+\begin_inset Formula $\langle P_{M}(x),y\rangle=\langle x,P_{M}(y)\rangle$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
+ y 
+\begin_inset Formula $\langle P_{M^{\bot}}(x),y\rangle=\langle x,P_{M^{\bot}}(y)\rangle$
+\end_inset
+
+.
 \end_layout
 
+\begin_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $x=x_{1}+x_{2}$
+\end_inset
+
+ e 
+\begin_inset Formula $y=y_{1}+y_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $x_{1},y_{1}\in M$
+\end_inset
+
+ y 
+\begin_inset Formula $x_{2},y_{2}\in M^{\bot}$
+\end_inset
+
+, 
+\begin_inset Formula $\langle P_{M}(x),y\rangle=\langle x_{1},y_{1}+y_{2}\rangle=\langle x_{1},y_{1}\rangle=\langle x_{1}+x_{2},y_{1}\rangle=\langle x,P_{M}(y)\rangle$
 \end_inset
 
+, y para 
+\begin_inset Formula $P_{M^{\bot}}$
+\end_inset
 
+ es análogo.
 \end_layout
 
+\end_deeper
 \begin_layout Enumerate
-\begin_inset Formula $A^{**}=A$
+\begin_inset Formula $M^{\bot\bot}=M$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
 \end_layout
 
+\begin_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $x\in M$
 \end_inset
 
+, para 
+\begin_inset Formula $y\in M^{\bot}$
+\end_inset
 
-\end_layout
+, 
+\begin_inset Formula $\langle y,x\rangle=\overline{\langle x,y\rangle}=0$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $(AC)^{*}=C^{*}A^{*}$
+, luego 
+\begin_inset Formula $x\in M^{\bot\bot}$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
+ Si 
+\begin_inset Formula $x\in M^{\bot\bot}\subseteq H$
+\end_inset
 
+, sean 
+\begin_inset Formula $y\in M$
 \end_inset
 
+ y 
+\begin_inset Formula $z\in M^{\bot}$
+\end_inset
 
-\end_layout
+ con 
+\begin_inset Formula $x=y+z$
+\end_inset
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $A$
+, 
+\begin_inset Formula $0=\langle x,z\rangle=\langle y,z\rangle+\langle z,z\rangle=\langle z,z\rangle=\Vert z\Vert^{2}$
 \end_inset
 
- es invertible, también lo es 
-\begin_inset Formula $A^{*}$
+, luego 
+\begin_inset Formula $z=0$
 \end_inset
 
  y 
-\begin_inset Formula $(A^{*})^{-1}=(A^{-1})^{*}$
+\begin_inset Formula $x\in M$
 \end_inset
 
 .
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Esto no es cierto si 
+\begin_inset Formula $M$
+\end_inset
+
+ no es cerrado ni si 
+\begin_inset Formula $H$
+\end_inset
+
+ no es completo.
 \begin_inset Note Note
 status open
 
@@ -1591,11 +1975,9 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-\begin_inset Formula $\Vert AA^{*}\Vert=\Vert A^{*}A\Vert=\Vert A\Vert^{2}$
-\end_inset
-
-.
+\begin_layout Standard
+Un espacio normado es de Hilbert si y sólo si cada subespacio cerrado tiene
+ un complementario topológico.
 \begin_inset Note Note
 status open
 
@@ -1608,168 +1990,206 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-\begin_inset Formula $\ker A=(\text{Im}A^{*})^{\bot}$
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
 \end_inset
 
- y 
-\begin_inset Formula $\ker A^{*}=(\text{Im}A)^{\bot}.$
+ es un espacio de Hilbert, 
+\begin_inset Formula $S\subseteq H$
 \end_inset
 
+ es total si y sólo si 
+\begin_inset Formula $S^{\bot}=0$
+\end_inset
 
-\begin_inset Note Note
-status open
+.
+\end_layout
 
-\begin_layout Plain Layout
-nproof
+\begin_layout Section
+Dual de un espacio de Hilbert
 \end_layout
 
+\begin_layout Standard
+
+\series bold
+Teorema de Riesz-Fréchet:
+\series default
+ Dados un espacio de Hilbert 
+\begin_inset Formula $H$
 \end_inset
 
+ y un operador 
+\begin_inset Formula $f:H\to\mathbb{K}$
+\end_inset
 
-\end_layout
+, 
+\begin_inset Formula $f$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $(\ker A)^{\bot}=\overline{\text{Im}A^{*}}$
+ es acotado si y sólo si existe 
+\begin_inset Formula $y\in H$
 \end_inset
 
- y 
-\begin_inset Formula $(\ker A^{*})^{\bot}=\overline{\text{Im}A}$
+ con 
+\begin_inset Formula $f=\langle\cdot,y\rangle$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $y$
+\end_inset
+
+ es único y 
+\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$
 \end_inset
 
 .
-\begin_inset Note Note
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
-nproof
-\end_layout
-
+\begin_inset Formula $\implies]$
 \end_inset
 
 
 \end_layout
 
-\begin_layout Standard
-Ejemplos:
-\end_layout
+\end_inset
 
-\begin_layout Enumerate
-En 
-\begin_inset Formula $\ell^{2}$
+Para la unicidad, si 
+\begin_inset Formula $f(x)=\langle x,y\rangle=\langle x,z\rangle$
 \end_inset
 
-, el adjunto de 
-\begin_inset Formula $S_{\text{r}}$
+ para todo 
+\begin_inset Formula $x\in H$
 \end_inset
 
- es 
-\begin_inset Formula $S_{\text{l}}$
+, 
+\begin_inset Formula $\langle x,y-z\rangle=0$
 \end_inset
 
- y viceversa.
-\begin_inset Note Note
-status open
+, luego 
+\begin_inset Formula $y-z\bot H$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ y, como 
+\begin_inset Formula $H^{\bot}=0$
+\end_inset
 
+, 
+\begin_inset Formula $y=z$
 \end_inset
 
+.
+ Para la existencia, si 
+\begin_inset Formula $f=0$
+\end_inset
 
-\end_layout
+ tomamos 
+\begin_inset Formula $y=0$
+\end_inset
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $H$
+, y en otro caso, 
+\begin_inset Formula $Y\coloneqq\ker f$
 \end_inset
 
- es un espacio de Hilbert y 
-\begin_inset Formula $K\in{\cal L}(H)$
+ es un subespacio cerrado de 
+\begin_inset Formula $H$
 \end_inset
 
- es un operador de rango finito dado por 
-\begin_inset Formula $K(x)=\sum_{i=1}^{n}\langle x,u_{i}\rangle v_{i}$
+ y por tanto 
+\begin_inset Formula $H=Y\oplus Y^{\bot}$
 \end_inset
 
-, su adjunto es de rango finito dado por 
-\begin_inset Formula $K^{*}(x)=\sum_{i=1}^{n}\langle x,v_{i}\rangle u_{i}$
+, con 
+\begin_inset Formula $\dim Y^{\bot}=\dim\text{Im}f=1$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
+ Sea entonces 
+\begin_inset Formula $z\in Y^{\bot}$
+\end_inset
 
+ unitario, la proyección ortogonal de un 
+\begin_inset Formula $x\in H$
 \end_inset
 
+ sobre 
+\begin_inset Formula $Y^{\bot}$
+\end_inset
 
-\end_layout
+ es 
+\begin_inset Formula $\langle x,z\rangle z$
+\end_inset
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $H$
+, luego 
+\begin_inset Formula $x-\langle x,z\rangle z\in Y$
 \end_inset
 
- es un espacio de Hilbert con base 
-\begin_inset Formula $(e_{i})_{i\in I}$
+ y
+\begin_inset Formula 
+\[
+f(x)=f(x-\langle x,z\rangle z+\langle x,z\rangle z)=f(\langle x,z\rangle z)=\langle x,z\rangle f(z)=\langle x,\overline{f(z)}z\rangle\eqqcolon\langle x,y\rangle.
+\]
+
 \end_inset
 
- y 
-\begin_inset Formula $A\in{\cal L}(H)$
+Para 
+\begin_inset Formula $x\in S_{H}$
 \end_inset
 
- es un operador diagonal con 
-\begin_inset Formula $A(e_{i})\coloneqq\lambda_{i}e_{i}$
+, por la desigualdad de Cauchy-Schwartz, 
+\begin_inset Formula $\Vert f(x)\Vert^{2}=|\langle x,y\rangle|^{2}\leq\langle x,x\rangle\langle y,y\rangle=\Vert y\Vert^{2}$
 \end_inset
 
- para ciertos 
-\begin_inset Formula $\lambda_{i}$
+, luego 
+\begin_inset Formula $\Vert f\Vert\leq\Vert y\Vert$
 \end_inset
 
-, entonces 
-\begin_inset Formula $A^{*}$
+, pero 
+\begin_inset Formula $f(\frac{y}{\Vert y\Vert})=\frac{f(y)}{\Vert y\Vert}=\frac{\Vert y\Vert^{2}}{\Vert y\Vert}=\Vert y\Vert$
 \end_inset
 
- es un operador diagonal con 
-\begin_inset Formula $A^{*}(e_{i})=\overline{\lambda_{i}}e_{i}$
+, luego 
+\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$
 \end_inset
 
 .
-\begin_inset Note Note
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
-nproof
-\end_layout
-
+\begin_inset Formula $\impliedby]$
 \end_inset
 
 
 \end_layout
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
 \end_inset
 
- es el operador multiplicación por 
-\begin_inset Formula $g\in L^{\infty}([a,b])$
-\end_inset
 
-, 
-\begin_inset Formula $K^{*}$
+\begin_inset Formula $f\coloneqq\langle\cdot,y\rangle$
 \end_inset
 
- es el operador multiplicación por 
-\begin_inset Formula $\overline{g}$
+ es lineal, y es continua por el argumento anterior que prueba que 
+\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$
 \end_inset
 
 .
+\end_layout
+
+\begin_layout Standard
+El teorema no es válido si 
+\begin_inset Formula $H$
+\end_inset
+
+ no es completo.
 \begin_inset Note Note
 status open
 
@@ -1782,32 +2202,35 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-Si 
+\begin_layout Standard
+Sean 
 \begin_inset Formula $H$
 \end_inset
 
- es un espacio de Hilbert separable con base hilbertiana 
-\begin_inset Formula $(e_{n})_{n\in I}$
+ un espacio de Hilbert y 
+\begin_inset Formula $T:H^{*}\to H$
 \end_inset
 
- y 
-\begin_inset Formula $A\in{\cal L}(H)$
+ que a cada 
+\begin_inset Formula $f$
 \end_inset
 
- se expresa en dicha base como 
-\begin_inset Formula $(a_{ij})\in\mathbb{K}^{I\times I}$
+ le asocia el 
+\begin_inset Formula $y$
 \end_inset
 
-, 
-\begin_inset Formula $A^{*}$
+ con 
+\begin_inset Formula $f=\langle\cdot,y\rangle$
 \end_inset
 
- se expresa en dicha base como 
-\begin_inset Formula $(\overline{a_{ji}})\in\mathbb{K}^{I\times I}$
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $T$
 \end_inset
 
-.
+ es biyectiva, isométrica y lineal conjugada.
 \begin_inset Note Note
 status open
 
@@ -1821,20 +2244,11 @@ nproof
 \end_layout
 
 \begin_layout Enumerate
-Si 
-\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
+\begin_inset Formula $H^{*}$
 \end_inset
 
- es el operador integral con núcleo 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
-\end_inset
-
-, 
-\begin_inset Formula $K^{*}$
-\end_inset
-
- es el operador integral con núcleo 
-\begin_inset Formula $k^{*}(t,s)\coloneqq\overline{k(s,t)}$
+ es un espacio de Hilbert con el producto escalar 
+\begin_inset Formula $\langle f,g\rangle^{*}\coloneqq\langle T(g),T(f)\rangle$
 \end_inset
 
 .
@@ -1851,23 +2265,14 @@ nproof
 \end_layout
 
 \begin_layout Enumerate
-Si 
-\begin_inset Formula $H$
+\begin_inset Formula $J:H\to H^{**}$
 \end_inset
 
- es un espacio de Hilbert, 
-\begin_inset Formula $M\leq H$
-\end_inset
-
- es cerrado e 
-\begin_inset Formula $\iota:M\hookrightarrow H$
-\end_inset
-
- es la inclusión, 
-\begin_inset Formula $\iota^{*}:H\to M$
+ dada por 
+\begin_inset Formula $J(x)(f)\coloneqq f(x)$
 \end_inset
 
- es la proyección ortogonal.
+ es un isomorfismo algebraico isométrico.
 \begin_inset Note Note
 status open
 
@@ -1881,63 +2286,103 @@ nproof
 \end_layout
 
 \begin_layout Standard
-En general el adjunto no existe en espacios prehilbertianos.
- Por ejemplo, 
-\begin_inset Formula $T:c_{00}\to c_{00}$
+Dado un un 
+\begin_inset Formula $\mathbb{K}$
 \end_inset
 
- dado por 
-\begin_inset Formula $T(x)\coloneqq\sum_{n\geq1}\frac{x_{n}}{n}(1,0,\dots)$
+-espacio vectorial 
+\begin_inset Formula $X$
 \end_inset
 
- no tiene adjunto en 
-\begin_inset Formula $(c_{00},\langle\cdot,\cdot\rangle_{2})$
+, 
+\begin_inset Formula $B:X\times X\to\mathbb{K}$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+ es 
+\series bold
+bilineal
+\series default
+ si las 
+\begin_inset Formula $B(\cdot,y)$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ y 
+\begin_inset Formula $B(x,\cdot)$
+\end_inset
 
+ son lineales, 
+\series bold
+sesquilineal
+\series default
+ si las 
+\begin_inset Formula $B(\cdot,y)$
 \end_inset
 
+ son lineales y las 
+\begin_inset Formula $B(x,\cdot)$
+\end_inset
 
-\end_layout
+ son lineales conjugadas, 
+\series bold
+simétrica
+\series default
+ si 
+\begin_inset Formula $B(x,y)\equiv B(y,x)$
+\end_inset
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $H$
+ y 
+\series bold
+positiva
+\series default
+ si 
+\begin_inset Formula $\forall x\in X,B(x,x)\geq0$
 \end_inset
 
- es un espacio de Hilbert, 
-\begin_inset Formula $A\in{\cal L}(H)$
+.
+ Si además 
+\begin_inset Formula $X$
+\end_inset
+
+ es normado, 
+\begin_inset Formula $B$
 \end_inset
 
  es 
 \series bold
-autoadjunto
+acotada
 \series default
- o 
+ si 
+\begin_inset Formula $\exists M>0:\forall x,y\in X,|B(x,y)|\leq M\Vert x\Vert\Vert y\Vert$
+\end_inset
+
+, y es 
 \series bold
-hermitiano
+fuertemente positiva
 \series default
  si 
-\begin_inset Formula $A^{*}=A$
+\begin_inset Formula $\exists c>0:\forall x\in X,B(x,x)\geq c\Vert x\Vert^{2}$
 \end_inset
 
 .
- Si 
-\begin_inset Formula $A,B\in{\cal L}(H)$
+ 
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $B$
 \end_inset
 
- son autoadjuntos:
-\end_layout
+ es bilineal o sesquilineal, es acotada si y sólo si es continua, y para
+ todo 
+\begin_inset Formula $x$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $\Vert A\Vert=\sup_{x\in\overline{B_{H}}}|\langle Ax,x\rangle|=\sup_{x\in S_{H}}|\langle Ax,x\rangle|$
+ e 
+\begin_inset Formula $y$
+\end_inset
+
+ es 
+\begin_inset Formula $2B(x,x)+2B(y,y)=B(x+y,x+y)+B(x-y,x-y)$
 \end_inset
 
 .
@@ -1953,593 +2398,843 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-Los valores propios de 
-\begin_inset Formula $A$
+\begin_layout Standard
+
+\series bold
+Teorema de Lax-Milgram:
+\series default
+ Sean 
+\begin_inset Formula $H$
 \end_inset
 
- son reales.
-\begin_inset Note Note
-status open
+ un espacio de Hilbert y 
+\begin_inset Formula $B$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ una 
+\begin_inset Formula $H$
+\end_inset
 
+-forma sesquilineal acotada y fuertemente positiva, existe un único isomorfismo
+ de espacios de Hilbert 
+\begin_inset Formula $T:H\to H$
 \end_inset
 
+ tal que 
+\begin_inset Formula $\forall x,y\in H,B(x,y)=\langle x,T(y)\rangle$
+\end_inset
 
-\end_layout
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sea
+\begin_inset Formula 
+\[
+Y\coloneqq\{y\in H\mid\exists z\in H:\langle\cdot,y\rangle=B(\cdot,z)\},
+\]
 
-\begin_layout Enumerate
-\begin_inset Formula $\forall x\in H,\langle Ax,x\rangle=0\implies A=0$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+\begin_inset Formula $0\in Y$
+\end_inset
 
+ tomando 
+\begin_inset Formula $z=0$
 \end_inset
 
+ y 
+\begin_inset Formula $z$
+\end_inset
 
-\end_layout
+ está unívocamente determinado por 
+\begin_inset Formula $y$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $H=\ker A\oplus\overline{\text{Im}A}$
+, ya que si 
+\begin_inset Formula $\langle\cdot,y\rangle=B(\cdot,z)=B(\cdot,z')$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+ entonces 
+\begin_inset Formula $B(\cdot,z-z')=0$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ y en particular 
+\begin_inset Formula $0=B(z-z',z-z')\geq c\Vert z-z'\Vert^{2}$
+\end_inset
 
+ para cierto 
+\begin_inset Formula $c>0$
 \end_inset
 
+ por ser 
+\begin_inset Formula $B$
+\end_inset
 
-\end_layout
+ fuertemente positiva, luego 
+\begin_inset Formula $z=z'$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $A+B$
+.
+ Como 
+\begin_inset Formula $\langle\cdot,\cdot\rangle$
 \end_inset
 
- es autoadjunto, y 
-\begin_inset Formula $AB$
+ y 
+\begin_inset Formula $B$
 \end_inset
 
- lo es si y sólo si 
-\begin_inset Formula $AB=BA$
+ son sesquilineales, 
+\begin_inset Formula $Y$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+ es un espacio vectorial y 
+\begin_inset Formula $S:Y\to H$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ que a cada 
+\begin_inset Formula $y$
+\end_inset
 
+ le asocia el 
+\begin_inset Formula $z$
 \end_inset
 
+ con 
+\begin_inset Formula $\langle\cdot,y\rangle=B(\cdot,z)$
+\end_inset
 
-\end_layout
+ es lineal.
+ Entonces, para 
+\begin_inset Formula $y\in S_{Y}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+c\Vert S(y)\Vert^{2}\leq B(S(y),S(y))=\langle S(y),y\rangle\in\mathbb{R}^{+},
+\]
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $_{\mathbb{C}}H$
 \end_inset
 
- es un espacio de Hilbert y 
-\begin_inset Formula $A\in{\cal L}(H)$
+pero por la desigualdad de Cauchy-Schwartz, 
+\begin_inset Formula $\langle S(y),y\rangle^{2}=|\langle S(y),y\rangle|^{2}\leq\Vert S(y)\Vert^{2}\Vert y\Vert^{2}$
 \end_inset
 
-:
-\end_layout
+, luego 
+\begin_inset Formula $c\Vert S(y)\Vert^{2}\leq\langle S(y),y\rangle\leq\Vert S(y)\Vert\Vert y\Vert=\Vert S(y)\Vert$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $A$
+ y 
+\begin_inset Formula $\Vert S(y)\Vert\leq\frac{1}{c}$
 \end_inset
 
- es autoadjunto si y sólo si 
-\begin_inset Formula $\forall x\in H,\langle Ax,x\rangle\in\mathbb{R}$
+, con lo que 
+\begin_inset Formula $S$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+ es continua.
+ Entonces, si 
+\begin_inset Formula $\{y_{n}\}_{n}\subseteq Y$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ y existe 
+\begin_inset Formula $\lim_{n}y_{n}\eqqcolon y\in H$
+\end_inset
 
+, por continuidad de 
+\begin_inset Formula $S$
 \end_inset
 
+ y de 
+\begin_inset Formula $B$
+\end_inset
 
-\end_layout
+,
+\begin_inset Formula 
+\[
+\langle x,y\rangle=\lim_{n}\langle x,y_{n}\rangle=\lim_{n}B(x,S(y_{n}))=B(x,S(y)),
+\]
 
-\begin_layout Enumerate
+\end_inset
 
-\backslash
-Existen únicos 
-\begin_inset Formula $\text{Re}A,\text{Im}A\in{\cal L}(H)$
+luego 
+\begin_inset Formula $y\in Y$
 \end_inset
 
- autoadjuntos, la 
-\series bold
-parte real
-\series default
- y la 
-\series bold
-imaginaria
-\series default
- de 
-\begin_inset Formula $A$
+ e 
+\begin_inset Formula $Y$
 \end_inset
 
-, con 
-\begin_inset Formula $A=\text{Re}A+\text{i}\text{Im}A$
+ es cerrado.
+ Entonces, si 
+\begin_inset Formula $z\in Y^{\bot}$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+, como 
+\begin_inset Formula $B(\cdot,z):H\to\mathbb{K}$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ es continua, por el teorema de Riesz-Fréchet existe 
+\begin_inset Formula $w\in H$
+\end_inset
 
+ con 
+\begin_inset Formula $B(\cdot,z)=\langle\cdot,w\rangle$
 \end_inset
 
+, luego 
+\begin_inset Formula $w\in Y$
+\end_inset
 
-\end_layout
+, pero entonces 
+\begin_inset Formula $B(z,z)=\langle z,w\rangle=0$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $\llbracket A\rrbracket\coloneqq\sup_{x\in S_{H}}|\langle Ax,x\rangle|$
+ y, por ser 
+\begin_inset Formula $B$
 \end_inset
 
- es una norma en 
-\begin_inset Formula ${\cal L}(H)$
+ fuertemente positiva, 
+\begin_inset Formula $z=0$
 \end_inset
 
- equivalente a la usual.
-\end_layout
+, luego 
+\begin_inset Formula $Y^{\bot}=0$
+\end_inset
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $H$
+ e 
+\begin_inset Formula $Y=H$
 \end_inset
 
- es un espacio de Hilbert con base 
-\begin_inset Formula $(e_{i})_{i\in I}$
+.
+ Para 
+\begin_inset Formula $z\in H$
 \end_inset
 
-:
-\end_layout
+, como 
+\begin_inset Formula $B(\cdot,z)$
+\end_inset
 
-\begin_layout Enumerate
-El operador diagonal 
-\begin_inset Formula $T\in{\cal L}(H)$
+ es continua, existe 
+\begin_inset Formula $w\in H$
 \end_inset
 
  con 
-\begin_inset Formula $T(e_{i})\eqqcolon\lambda_{i}e_{i}$
+\begin_inset Formula $B(\cdot z)=\langle\cdot,w\rangle$
 \end_inset
 
- es autoadjunto si y sólo si 
-\begin_inset Formula $\{\lambda_{i}\}_{i\in I}\subseteq\mathbb{R}$
+ y por tanto 
+\begin_inset Formula $z=S(w)$
 \end_inset
 
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $H$
+, luego 
+\begin_inset Formula $S$
 \end_inset
 
- es separable y 
-\begin_inset Formula $A\in{\cal L}(H)$
+ es suprayectiva.
+ Si 
+\begin_inset Formula $S(y)=0$
 \end_inset
 
- se representa respecto a la base como la matriz 
-\begin_inset Formula $(a_{ij})\in\mathbb{K}^{I\times I}$
+, para 
+\begin_inset Formula $x\in H$
 \end_inset
 
 , 
-\begin_inset Formula $A$
+\begin_inset Formula $\langle x,y\rangle=B(x,S(y))=0$
 \end_inset
 
- es autoadjunto si y sólo si 
-\begin_inset Formula $\forall i,j\in I,a_{ij}=\overline{a_{ji}}$
+ y por tanto 
+\begin_inset Formula $y=0$
 \end_inset
 
-.
-\end_layout
-
-\begin_layout Enumerate
-El operador multiplicación por 
-\begin_inset Formula $g\in L^{\infty}([a,b])$
+, luego 
+\begin_inset Formula $S$
 \end_inset
 
- en 
-\begin_inset Formula $L^{2}([a,b])$
+ es inyectiva.
+ Por tanto 
+\begin_inset Formula $S$
 \end_inset
 
- es autoadjunto si y sólo si 
-\begin_inset Formula $g(t)$
+ es biyectiva y 
+\begin_inset Formula $T\coloneqq S^{-1}$
 \end_inset
 
- es real para casi todo 
-\begin_inset Formula $t\in[a,b]$
+ cumple 
+\begin_inset Formula $\langle x,T(y)\rangle=B(x,y)$
 \end_inset
 
 .
-\end_layout
+ Además, para 
+\begin_inset Formula $y\in S_{H}$
+\end_inset
 
-\begin_layout Enumerate
-El operador integral con núcleo 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+, 
+\begin_inset Formula $\Vert T(y)\Vert^{2}=\langle T(y),T(y)\rangle=B(T(y),y)\leq M\Vert T(y)\Vert\Vert y\Vert=M\Vert T(y)\Vert$
 \end_inset
 
- en 
-\begin_inset Formula $L^{2}([a,b])$
+, siendo 
+\begin_inset Formula $M$
 \end_inset
 
- es autoadjunto si y sólo si 
-\begin_inset Formula $k(t,s)=\overline{k(s,t)}$
+ una cota de 
+\begin_inset Formula $B$
 \end_inset
 
- para casi todo 
-\begin_inset Formula $(s,t)\in[a,b]\times[a,b]$
+, de donde 
+\begin_inset Formula $\Vert T\Vert\leq M$
 \end_inset
 
-.
+ y, como 
+\begin_inset Formula $\Vert T^{-1}\Vert=\Vert S\Vert\leq\frac{1}{c}$
+\end_inset
+
+, 
+\begin_inset Formula $T$
+\end_inset
+
+ es un isomorfismo topológico isométrico.
 \end_layout
 
-\begin_layout Enumerate
-Una proyección ortogonal 
-\begin_inset Formula $P:H\to H$
+\begin_layout Standard
+En particular, dado un espacio vectorial 
+\begin_inset Formula $H$
 \end_inset
 
- sobre un subespacio cerrado es autoadjunto.
-\begin_inset Note Note
-status open
+ con dos productos escalares 
+\begin_inset Formula $\langle\cdot,\cdot\rangle_{1}$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ y 
+\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$
+\end_inset
 
+ equivalentes que hacen a 
+\begin_inset Formula $H$
 \end_inset
 
+ completo, existe un isomorfismo 
+\begin_inset Formula $T:H\to H$
+\end_inset
 
+ de espacios de Hilbert con 
+\begin_inset Formula $\langle x,y\rangle_{1}=\langle x,T(y)\rangle_{2}$
+\end_inset
+
+.
 \end_layout
 
 \begin_layout Standard
-Si 
-\begin_inset Formula $H$
+Dado un espacio medible 
+\begin_inset Formula $(\Omega,\Sigma)$
 \end_inset
 
- es un espacio de Hilbert, 
-\begin_inset Formula $A\in{\cal L}(H)$
+ con medidas 
+\begin_inset Formula $\mu$
+\end_inset
+
+ y 
+\begin_inset Formula $\nu$
+\end_inset
+
+, 
+\begin_inset Formula $\nu$
 \end_inset
 
  es 
 \series bold
-normal
+absolutamente continua
 \series default
- si 
-\begin_inset Formula $AA^{*}=A^{*}A$
+ respecto de 
+\begin_inset Formula $\mu$
 \end_inset
 
-, si y sólo si 
-\begin_inset Formula $\forall x,y\in H,\langle Ax,Ay\rangle=\langle A^{*}x,A^{*}y\rangle$
+ si 
+\begin_inset Formula $\forall A\in\Sigma,(\mu(A)=0\implies\nu(A)=0)$
 \end_inset
 
-, si y sólo si 
-\begin_inset Formula $\forall x\in H,\Vert Ax\Vert=\Vert A^{*}x\Vert$
+, y es 
+\series bold
+finita
+\series default
+ si 
+\begin_inset Formula $\nu(\Omega)<\infty$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
+ 
+\series bold
+Teorema de Radon-Nicodym:
+\series default
+ Si 
+\begin_inset Formula $(\Omega,\Sigma)$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ es un espacio medible con medidas finitas 
+\begin_inset Formula $\mu$
+\end_inset
 
+ y 
+\begin_inset Formula $\nu$
 \end_inset
 
+ siendo 
+\begin_inset Formula $\nu$
+\end_inset
 
-\end_layout
+ absolutamente continua respecto de 
+\begin_inset Formula $\mu$
+\end_inset
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $H$
+, existe 
+\begin_inset Formula $g:\Omega\to[0,+\infty]$
 \end_inset
 
- es un espacio de Hilbert complejo, 
-\begin_inset Formula $A\in{\cal L}(H)$
+ 
+\begin_inset Formula $\mu$
 \end_inset
 
- es normal si y sólo si 
-\begin_inset Formula $\text{Re}A\circ\text{Im}A=\text{Im}A\circ\text{Re}A$
+-integrable tal que
+\begin_inset Formula 
+\[
+\forall A\in\Sigma,\nu(A)=\int_{A}g\dif\mu.
+\]
+
 \end_inset
 
-.
-\begin_inset Note Note
-status open
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $\sigma\coloneqq\mu+\nu$
+\end_inset
 
+ es una medida finita en 
+\begin_inset Formula $X$
 \end_inset
 
+ tal que 
+\begin_inset Formula $\forall A\in\Sigma,(\sigma(A)=0\iff\mu(A)=0)$
+\end_inset
 
-\end_layout
+, y la función lineal entre espacios de Hilbert 
+\begin_inset Formula $T:L^{2}(\Omega,\Sigma,\sigma)\to\mathbb{R}$
+\end_inset
 
-\begin_layout Enumerate
-Todo operador diagonal es normal.
-\end_layout
+ dada por
+\begin_inset Formula 
+\[
+Tu\coloneqq\int_{\Omega}u\dif\mu
+\]
 
-\begin_layout Enumerate
-El operador integral sobre 
-\begin_inset Formula $L^{2}([a,b])$
 \end_inset
 
- con núcleo 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+está bien definida y es continua porque, si 
+\begin_inset Formula $\Vert u\Vert_{L^{2}(\Omega,\Sigma,\sigma)}=1$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{align*}
+|Tu| & =\left|\int_{\Omega}u\dif\mu\right|\leq\int_{\Omega}|u|\dif\mu\leq\sqrt{\int_{\Omega}|u|^{2}\dif\mu}+\sqrt{\int_{\Omega}\dif\mu}\leq\\
+ & \leq\sqrt{\int_{\Omega}|u|^{2}\dif\mu+\int_{\Omega}|u|^{2}\dif\nu}+\sqrt{\int_{\Omega}\dif\mu+\int_{\Omega}\dif\nu}=1+\sqrt{\sigma(X)}.
+\end{align*}
+
+\end_inset
+
+Por el teorema de representación de Riesz, existe 
+\begin_inset Formula $f\in L^{2}(\Omega,\Sigma,\sigma)$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $u\in L^{2}(\Omega,\Sigma,\sigma)$
 \end_inset
 
- es normal si y sólo si
+,
 \begin_inset Formula 
 \[
-\int_{a}^{b}\overline{k(s,t)}k(s,x)\dif s=\int_{a}^{b}k(t,s)\overline{k(x,s)}\dif s
+Tu=\int_{\Omega}u\dif\mu=\int_{\Omega}uf\dif\sigma,
 \]
 
 \end_inset
 
-para casi todo 
-\begin_inset Formula $(t,x)\in[a,b]\times[a,b]$
+pero esta igualdad se da para cuando 
+\begin_inset Formula $u=\chi_{A}$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+ para cualquier 
+\begin_inset Formula $A\in{\cal F}$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ y por linealidad para cualquier función 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+-medible simple, y por el teorema de convergencia dominada también se da
+ para cualquier función 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+-medible no negativa en casi todo punto.
+ Además, para 
+\begin_inset Formula $A\in\Sigma$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\mu(A)=\int_{\Omega}\chi_{A}f\dif\sigma=\int_{A}f\dif\sigma,
+\]
+
+\end_inset
+
+de modo que 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+-medible y, haciendo 
+\begin_inset Formula $A=\{x\mid f(x)\leq0\}$
+\end_inset
+
+ o 
+\begin_inset Formula $A=\{x\mid f(x)>1\}$
+\end_inset
+
+, vemos que 
+\begin_inset Formula $f(\omega)\in(0,1]$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $\omega\in\Omega$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $\frac{1}{g}$
+\end_inset
+
+ es 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+-medible no negativa en casi todo punto y, en casi todo punto, 
+\begin_inset Formula $\frac{1}{f}f=1$
+\end_inset
+
+, con lo que para 
+\begin_inset Formula $A\in\Sigma$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\int_{A}\frac{1}{f}\dif\mu=\int_{A}\dif\sigma\implies\nu(A)=\sigma(A)-\mu(A)=\int_{A}\left(\frac{1}{f}-1\right)\dif\mu\eqqcolon\int_{A}g\dif\mu.
+\]
 
 \end_inset
 
 
 \end_layout
 
+\begin_layout Section
+Problemas variacionales cuadráticos
+\end_layout
+
 \begin_layout Standard
-Una 
+
 \series bold
-proyección
+Teorema principal de los problemas variacionales cuadráticos:
 \series default
- en un espacio normado 
-\begin_inset Formula $X$
+ Sean 
+\begin_inset Formula $H$
 \end_inset
 
- es un operador 
-\begin_inset Formula $X\to X$
+ un 
+\begin_inset Formula $\mathbb{R}$
 \end_inset
 
- idempotente.
- Si 
-\begin_inset Formula $H$
+-espacio de Hilbert, 
+\begin_inset Formula $B$
 \end_inset
 
- es un espacio de Hilbert y 
-\begin_inset Formula $P$
+ una 
+\begin_inset Formula $H$
 \end_inset
 
- es una proyección continua no nula en 
-\begin_inset Formula $X$
+-forma bilineal simétrica, acotada y fuertemente positiva, 
+\begin_inset Formula $b$
 \end_inset
 
-, 
-\begin_inset Formula $P$
+ una 
+\begin_inset Formula $H$
 \end_inset
 
- es una proyección ortogonal si y sólo si 
-\begin_inset Formula $\Vert P\Vert=1$
+-forma lineal continua y 
+\begin_inset Formula $F:H\to\mathbb{R}$
 \end_inset
 
-, si y sólo si 
-\begin_inset Formula $\text{Im}P=(\ker P)^{\bot}$
+ dada por
+\begin_inset Formula 
+\[
+F(x)\coloneqq\frac{1}{2}B(x,x)-b(x),
+\]
+
 \end_inset
 
-, si y sólo si 
-\begin_inset Formula $\ker P=(\text{Im}P)^{\bot}$
+entonces:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $w\in H$
 \end_inset
 
-, si y sólo si 
-\begin_inset Formula $P$
+, 
+\begin_inset Formula $F$
 \end_inset
 
- es autoadjunto, si y sólo si es normal, si y sólo si 
-\begin_inset Formula $\forall x\in H,\langle Px,x\rangle=\Vert Px\Vert^{2}$
+ alcanza su mínimo en 
+\begin_inset Formula $w$
 \end_inset
 
-, si y sólo si 
-\begin_inset Formula $\forall x\in H,\langle Px,x\rangle\geq0$
+ si y sólo si 
+\begin_inset Formula $\forall y\in H,B(w,y)=b(y)$
 \end_inset
 
 .
-\begin_inset Note Note
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
-nproof
+\begin_inset Formula $\implies]$
+\end_inset
+
+
 \end_layout
 
 \end_inset
 
+Fijado 
+\begin_inset Formula $y\in H$
+\end_inset
+
+, para 
+\begin_inset Formula $t\in\mathbb{R}$
+\end_inset
 
-\end_layout
 
-\begin_layout Standard
-Existen proyecciones no ortogonales, como 
-\begin_inset Formula $p:\mathbb{R}^{2}\to\mathbb{R}^{2}$
+\begin_inset Formula 
+\begin{align*}
+F(w+ty) & =\frac{1}{2}B(w+ty,w+ty)-b(w+ty)=\\
+ & =\frac{1}{2}(B(w,w)+2tB(w,y)+t^{2}B(y,y))-b(w)-tb(y)=\\
+ & =F(w)+t(B(w,y)-b(y))+\frac{1}{2}t^{2}B(y,y),
+\end{align*}
+
+\end_inset
+
+pero por hipótesis 
+\begin_inset Formula $F(w)\leq F(w+ty)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $t\in\mathbb{R}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\varphi:\mathbb{R}\to\mathbb{R}$
 \end_inset
 
  dada por 
-\begin_inset Formula $p(x,y)\coloneqq(x+y,0)$
+\begin_inset Formula $\varphi(t)\coloneqq F(w+ty)$
+\end_inset
+
+ tiene un mínimo en 
+\begin_inset Formula $t=0$
+\end_inset
+
+ y 
+\begin_inset Formula $0=\varphi'(0)=B(w,y)-b(y)$
 \end_inset
 
 .
 \end_layout
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $H$
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
 \end_inset
 
- es un 
-\begin_inset Formula $\mathbb{K}$
+
+\end_layout
+
 \end_inset
 
--espacio de Hilbert, 
-\begin_inset Formula $T\in{\cal L}(H)$
+Para 
+\begin_inset Formula $y\in H$
 \end_inset
 
  y 
-\begin_inset Formula $\lambda\in\mathbb{K}$
+\begin_inset Formula $t\in\mathbb{R}$
 \end_inset
 
 , 
-\begin_inset Formula $\lambda\in\sigma(T)\iff\overline{\lambda}\in\sigma(T^{*})$
+\begin_inset Formula 
+\[
+F(w+ty)=F(w)+\cancel{t(B(w,y)-b(y))}^{=0}+\frac{1}{2}t^{2}B(y,y)\geq F(w).
+\]
+
 \end_inset
 
-.
-\begin_inset Note Note
-status open
 
-\begin_layout Plain Layout
-nproof
 \end_layout
 
+\end_deeper
+\begin_layout Enumerate
+Existe un único 
+\begin_inset Formula $w\in H$
 \end_inset
 
+ en el que 
+\begin_inset Formula $F$
+\end_inset
 
+ alcanza su mínimo.
 \end_layout
 
+\begin_deeper
 \begin_layout Standard
-Si 
-\begin_inset Formula $T\in{\cal L}(H)$
+Como 
+\begin_inset Formula $B$
 \end_inset
 
- es normal:
-\end_layout
+ es bilineal, simétrica y fuertemente positiva, es un producto escalar sobre
+ 
+\begin_inset Formula $H$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $\forall\lambda\in\mathbb{C}$
+, y como existen 
+\begin_inset Formula $c,M>0$
 \end_inset
 
-, 
-\begin_inset Formula $\ker(T-\lambda1_{H})=\ker(T^{*}-\overline{\lambda}1_{H})$
+ con 
+\begin_inset Formula $c\Vert x\Vert^{2}\leq B(x,x)\leq M\Vert x\Vert^{2}$
 \end_inset
 
-.
-\end_layout
+, el producto escalar 
+\begin_inset Formula $B$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $\forall\lambda,\mu\in\mathbb{C},(\lambda\neq\mu\implies\ker(T-\lambda1_{H})\bot\ker(T-\mu1_{H}))$
+ es equivalente al de 
+\begin_inset Formula $H$
 \end_inset
 
-.
-\end_layout
+, luego 
+\begin_inset Formula $b$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $\ker(T-\lambda1_{H})$
+ es continua con el producto escalar 
+\begin_inset Formula $B$
 \end_inset
 
- y 
-\begin_inset Formula $\ker(T-\lambda1_{H})^{\bot}$
+ y por el teorema de Riesz-Fréchet existe un único 
+\begin_inset Formula $w\in H$
 \end_inset
 
- son 
-\begin_inset Formula $T$
+ con 
+\begin_inset Formula $b=B(\cdot,w)=B(w,\cdot)$
 \end_inset
 
--invariantes.
+, que es la condición del primer apartado.
 \end_layout
 
+\end_deeper
 \begin_layout Section
-Operadores compactos
+Convolución y aproximación de funciones
 \end_layout
 
 \begin_layout Standard
-Dado un espacio topológico 
-\begin_inset Formula $X$
+Dado un abierto 
+\begin_inset Formula $\Omega\subseteq\mathbb{R}^{n}$
 \end_inset
 
 , 
-\begin_inset Formula $Y\subseteq X$
+\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$
 \end_inset
 
  es 
 \series bold
-relativamente compacto
+localmente integrable
 \series default
- en 
-\begin_inset Formula $X$
+ si 
+\begin_inset Formula $|f|$
 \end_inset
 
- si su clausura en 
-\begin_inset Formula $X$
+ es integrable en todo compacto 
+\begin_inset Formula $K\subseteq\Omega$
 \end_inset
 
- es compacta.
- Sean 
-\begin_inset Formula $X$
+.
+ Dadas dos funciones localmente integrables 
+\begin_inset Formula $f,g:\mathbb{R}^{n}\to\mathbb{R}$
 \end_inset
 
- e 
-\begin_inset Formula $Y$
+, definimos su 
+\series bold
+producto de convolución
+\series default
+ como 
+\begin_inset Formula $(f*g):D\to\mathbb{R}$
 \end_inset
 
- espacios normados, una función lineal 
-\begin_inset Formula $T:X\to Y$
-\end_inset
+ dada por
+\begin_inset Formula 
+\[
+(f*g)(a)\coloneqq\int_{\mathbb{R}^{n}}f(x)g(a-x)\dif x,
+\]
 
- es 
-\series bold
-compacta
-\series default
- si 
-\begin_inset Formula $T(B_{X})$
 \end_inset
 
- es relativamente compacta en 
-\begin_inset Formula $Y$
+donde 
+\begin_inset Formula $D\coloneqq\{a\in\mathbb{R}^{n}\mid x\mapsto f(x)g(a-x)\text{ integrable}\}$
 \end_inset
 
-, si y sólo si para cada sucesión acotada 
-\begin_inset Formula $\{x_{n}\}_{n}\subseteq X$
+.
+ Si 
+\begin_inset Formula $f,g\in L^{2}(\mathbb{R}^{n})$
 \end_inset
 
 , 
-\begin_inset Formula $(Tx_{n})_{n}$
+\begin_inset Formula $f*g$
 \end_inset
 
- posee una subsucesión convergente, si y sólo si esto se cumple cuando cada
- 
-\begin_inset Formula $\Vert x_{n}\Vert=1$
+ está definida en todo 
+\begin_inset Formula $\mathbb{R}^{n}$
 \end_inset
 
-.
+ y es continua y uniformemente acotada con
+\begin_inset Formula 
+\[
+\Vert f*g\Vert_{\infty}\leq\Vert f\Vert_{2}\Vert g\Vert_{2}.
+\]
+
+\end_inset
+
+
 \begin_inset Note Note
 status open
 
@@ -2549,11 +3244,15 @@ nproof
 
 \end_inset
 
+El producto de convolución es conmutativo, y si 
+\begin_inset Formula $f*g$
+\end_inset
 
-\end_layout
+ está definida en casi todo punto, 
+\begin_inset Formula $\text{sop}(f*g)\subseteq\overline{\text{sop}(f)+\text{sop}(g)}$
+\end_inset
 
-\begin_layout Enumerate
-Los operadores de rango finito son compactos.
+.
 \begin_inset Note Note
 status open
 
@@ -2566,8 +3265,40 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-El operador identidad en un espacio de dimensión infinita nunca es compacto.
+\begin_layout Standard
+Una 
+\series bold
+sucesión de Dirac
+\series default
+ es una sucesión 
+\begin_inset Formula $(K_{m}:\mathbb{R}^{n}\to\mathbb{R}^{\geq0})_{m}$
+\end_inset
+
+ de funciones continuas con
+\begin_inset Formula 
+\[
+\int_{\mathbb{R}^{n}}K_{n}=1
+\]
+
+\end_inset
+
+y tal que
+\begin_inset Formula 
+\[
+\forall\varepsilon,\delta>0,\exists n_{0}:\forall n\geq n_{0},\int_{\mathbb{R}^{n}\setminus B(0,\delta)}K_{n}(x)\dif x<\varepsilon.
+\]
+
+\end_inset
+
+Por ejemplo, si 
+\begin_inset Formula $K:\mathbb{R}^{n}\to\mathbb{R}$
+\end_inset
+
+ es continua, no negativa, con soporte compacto e integral 1, entonces 
+\begin_inset Formula $(x\mapsto m^{n}K(mx))_{m\geq1}$
+\end_inset
+
+ es una sucesión de Dirac.
 \begin_inset Note Note
 status open
 
@@ -2581,19 +3312,44 @@ nproof
 \end_layout
 
 \begin_layout Standard
-Llamamos 
-\begin_inset Formula ${\cal K}(X,Y)$
+Las sucesiones de Dirac aproximan la 
+\series bold
+delta de Dirac
+\series default
+, una 
+\begin_inset Quotes cld
 \end_inset
 
- al subespacio vectorial de 
-\begin_inset Formula ${\cal L}(X,Y)$
+función extendida
+\begin_inset Quotes crd
 \end_inset
 
- de los operadores compactos, que es cerrado si 
-\begin_inset Formula $Y$
+ con integral 1 que vale 0 en todo punto salvo en el origen en que el valor
+ es infinito.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$
+\end_inset
+
+ es continua y acotada, la sucesión 
+\begin_inset Formula $(f*K_{m})_{m}$
+\end_inset
+
+ tiende uniformemente a 
+\begin_inset Formula $f$
+\end_inset
+
+ sobre subconjuntos compactos de 
+\begin_inset Formula $\mathbb{R}^{n}$
 \end_inset
 
- es de Banach.
+.
 \begin_inset Note Note
 status open
 
@@ -2608,27 +3364,43 @@ nproof
 
 \begin_layout Standard
 Si 
-\begin_inset Formula $A\in{\cal L}(X,Y)$
+\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$
+\end_inset
+
+ es localmente integrable y 
+\begin_inset Formula $g\in{\cal D}^{k}(\mathbb{R}^{n})$
 \end_inset
 
 , 
-\begin_inset Formula $T\in{\cal K}(Y,Z)$
+\begin_inset Formula $f*g\in{\cal C}^{k}(\mathbb{R}^{n})$
 \end_inset
 
- y 
-\begin_inset Formula $B\in{\cal L}(Z,W)$
+ y para 
+\begin_inset Formula $\alpha\in\mathbb{N}^{n}$
 \end_inset
 
-, 
-\begin_inset Formula $BTA\in{\cal K}(X,W)$
+ con 
+\begin_inset Formula $\sum_{i}\alpha_{i}\leq k$
 \end_inset
 
-, y en particular 
-\begin_inset Formula ${\cal K}(X)\coloneqq{\cal K}(X,X)$
+ es
+\begin_inset Formula 
+\[
+\frac{\partial^{|\alpha|}(f*g)}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}=f*\left(\frac{\partial^{|\alpha|}g}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}\right),
+\]
+
 \end_inset
 
- es un ideal de 
-\begin_inset Formula ${\cal L}(X)$
+con lo que 
+\begin_inset Formula $f*g$
+\end_inset
+
+ es una regularización de 
+\begin_inset Formula $f$
+\end_inset
+
+ a través de una función suave 
+\begin_inset Formula $g$
 \end_inset
 
 .
@@ -2645,19 +3417,28 @@ nproof
 \end_layout
 
 \begin_layout Standard
-Si 
-\begin_inset Formula $T\in{\cal K}(X,Y)$
+Como 
+\series bold
+teorema
+\series default
+, dado un abierto 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
-:
-\end_layout
+, 
+\begin_inset Formula ${\cal D}(G)$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $\text{Im}T$
+ es denso en 
+\begin_inset Formula $(C_{c}(G),\Vert\cdot\Vert_{\infty})$
 \end_inset
 
- es un subespacio separable de 
-\begin_inset Formula $Y$
+ y en 
+\begin_inset Formula $L^{p}(G)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $p\in[1,\infty)$
 \end_inset
 
 .
@@ -2673,33 +3454,37 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $Y$
+\begin_layout Standard
+Para 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
- es de Hilbert, 
-\begin_inset Formula $\overline{\text{Im}T}$
+ abierto y 
+\begin_inset Formula $f\in L^{2}(G)$
 \end_inset
 
- es de dimensión infinita con base hilbertiana 
-\begin_inset Formula $(e_{n})_{n\in\mathbb{N}}$
+, si para todo 
+\begin_inset Formula $\psi\in{\cal D}(G)$
 \end_inset
 
- y, para 
-\begin_inset Formula $n\in\mathbb{N}$
+ es
+\begin_inset Formula 
+\[
+\int_{G}f\psi=0
+\]
+
 \end_inset
 
-, 
-\begin_inset Formula $P_{n}\in{\cal L}(Y)$
+entonces 
+\begin_inset Formula $f=0$
 \end_inset
 
- es la proyección ortogonal sobre 
-\begin_inset Formula $\text{span}\{e_{i}\}_{i\leq n}$
+ en casi todo punto, y en particular, si 
+\begin_inset Formula $f$
 \end_inset
 
-, entonces 
-\begin_inset Formula $T=\lim_{n}P_{n}T\in{\cal L}(X,Y)$
+ es continua, 
+\begin_inset Formula $f=0$
 \end_inset
 
 .
@@ -2715,110 +3500,163 @@ nproof
 
 \end_layout
 
+\begin_layout Section
+Principio de Dirichlet
+\end_layout
+
 \begin_layout Standard
-Así, si 
-\begin_inset Formula $Y$
+Dado un abierto 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
- es de Hilbert, 
-\begin_inset Formula ${\cal K}(X,Y)$
+, 
+\begin_inset Formula $u\in{\cal D}^{2}(G)$
 \end_inset
 
- es la clausura en 
-\begin_inset Formula ${\cal L}(X,Y)$
+ es 
+\series bold
+armónica
+\series default
+ en 
+\begin_inset Formula $G$
 \end_inset
 
- del conjunto de operadores acotados de rango finito.
- Esto no es cierto cuando 
-\begin_inset Formula $Y$
+ si 
+\begin_inset Formula $\triangle u\coloneqq\nabla^{2}u=0$
 \end_inset
 
- es un espacio de Banach arbitrario.
-\begin_inset Note Note
-status open
+ en todo punto de 
+\begin_inset Formula $G$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+.
+ Dada 
+\begin_inset Formula $g\in{\cal C}(S_{\mathbb{C}})$
+\end_inset
 
+, el 
+\series bold
+problema de Dirichlet
+\series default
+ consiste en encontrar 
+\begin_inset Formula $u\in{\cal D}^{2}(\overline{B_{X}})$
 \end_inset
 
+ armónica con 
+\begin_inset Formula $u|_{S_{\mathbb{C}}}=g$
+\end_inset
 
-\end_layout
+.
+ Para un abierto 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
+\end_inset
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $G$
+, llamamos 
+\begin_inset Formula ${\cal C}^{m}(\overline{G})$
 \end_inset
 
- y 
-\begin_inset Formula $H$
+ al conjunto de funciones 
+\begin_inset Formula $u:\overline{G}\to\mathbb{R}$
 \end_inset
 
- son espacios de Hilbert, 
-\begin_inset Formula $T\in{\cal L}(G,H)$
+ con 
+\begin_inset Formula $u|_{G}\in{\cal C}^{m}(G)$
 \end_inset
 
- es compacto si y sólo si lo es 
-\begin_inset Formula $T^{*}$
+ para las que las derivadas parciales de orden 
+\begin_inset Formula $m$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+ de 
+\begin_inset Formula $u$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ en 
+\begin_inset Formula $G$
+\end_inset
 
+ admiten prolongación continua a 
+\begin_inset Formula $\overline{G}$
 \end_inset
 
- 
+.
+ Escribimos 
+\begin_inset Formula $\partial_{j}u\coloneqq\frac{\partial u}{\partial j}$
+\end_inset
+
+.
 \end_layout
 
 \begin_layout Standard
-Con esto:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{samepage}
 \end_layout
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $(e_{n})_{n\in\mathbb{N}}$
 \end_inset
 
- y 
-\begin_inset Formula $(f_{n})_{n\in\mathbb{N}}$
+Dados un abierto 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
- son bases hilbertianas respectivas de 
-\begin_inset Formula $G$
+ acotado y no vacío, 
+\begin_inset Formula $f:G\to\mathbb{R}$
 \end_inset
 
  y 
-\begin_inset Formula $H$
+\begin_inset Formula $g:\partial G\to\mathbb{R}$
+\end_inset
+
+, el 
+\series bold
+problema de valores frontera para la ecuación de Poisson
+\series default
+ consiste en encontrar 
+\begin_inset Formula $u:\overline{G}\to\mathbb{R}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $-\triangle u|_{G}=f$
 \end_inset
 
  y 
-\begin_inset Formula $T:G\to H$
+\begin_inset Formula $u|_{\partial G}=g$
 \end_inset
 
- es un operador diagonal dado por 
-\begin_inset Formula $Te_{n}\coloneqq\lambda_{n}f_{n}$
+, y el 
+\series bold
+problema generalizado de valores frontera
+\series default
+ consiste en encontrar 
+\begin_inset Formula $u:\overline{G}\to\mathbb{R}$
 \end_inset
 
-, 
-\begin_inset Formula $T$
+ con 
+\begin_inset Formula $u|_{\partial G}=g$
 \end_inset
 
- es compacto si y sólo si 
-\begin_inset Formula $\lim_{n}\lambda_{n}=0$
+ y
+\begin_inset Formula 
+\[
+\forall v\in{\cal D}(G),\int_{G}\sum_{j=1}^{n}\frac{\partial u}{\partial x_{j}}\frac{\partial v}{\partial x_{j}}\dif x\int_{G}fv.
+\]
+
 \end_inset
 
-.
-\begin_inset Note Note
+
+\begin_inset ERT
 status open
 
 \begin_layout Plain Layout
-nproof
+
+
+\backslash
+end{samepage}
 \end_layout
 
 \end_inset
@@ -2826,16 +3664,29 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-El operador multiplicación por 
-\begin_inset Formula $g\in L^{\infty}([a,b])$
+\begin_layout Standard
+Si 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
- es compacto si y sólo si 
-\begin_inset Formula $g=0$
+ es un abierto acotado no vacío, 
+\begin_inset Formula $f\in{\cal C}(\overline{G})$
 \end_inset
 
-.
+ y 
+\begin_inset Formula $g\in{\cal C}(\partial G)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Una 
+\begin_inset Formula $w\in{\cal C}^{2}(\overline{G})$
+\end_inset
+
+ es solución del problema de valores frontera para la ecuación de Poisson
+ y sólo si lo es del problema generalizado de valores frontera.
 \begin_inset Note Note
 status open
 
@@ -2850,42 +3701,61 @@ nproof
 
 \begin_layout Enumerate
 Si 
-\begin_inset Formula $G$
+\begin_inset Formula $w\in{\cal C}^{2}(\overline{G})$
 \end_inset
 
- y 
-\begin_inset Formula $H$
+ es solución del problema variacional consistente en encontrar el mínimo
+ de 
+\begin_inset Formula $F:\{u\in{\cal C}^{2}(\overline{G})\mid u|_{\partial G}=g\}\to\mathbb{R}$
 \end_inset
 
- son espacios de Hilbert de dimensión 
-\begin_inset Formula $\aleph_{0}$
-\end_inset
+ dada por
+\begin_inset Formula 
+\[
+F(u)\coloneqq\frac{1}{2}\int_{G}\sum_{j=1}^{n}(\partial_{j}u(x))^{2}\dif x-\int_{G}fu,
+\]
 
- y 
-\begin_inset Formula $T\in{\cal L}(G,H)$
 \end_inset
 
- se representa en ciertas bases de 
-\begin_inset Formula $G$
+entonces es solución de los dos problemas anteriores.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
- y 
-\begin_inset Formula $H$
+
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema de integración por partes en varias variables
+\series default
+ afirma que, si 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
- como 
-\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+ es un abierto, 
+\begin_inset Formula $u\in{\cal C}^{1}(G)$
 \end_inset
 
-, si 
-\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<\infty$
+ y 
+\begin_inset Formula $v\in{\cal D}(G)$
 \end_inset
 
-, 
-\begin_inset Formula $T$
+,
+\begin_inset Formula 
+\[
+\int_{G}u\partial_{j}v=-\int_{G}(\partial_{j}u)v.
+\]
+
 \end_inset
 
- es compacto.
+
 \begin_inset Note Note
 status open
 
@@ -2898,75 +3768,131 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-El operador integral 
-\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
+\begin_layout Standard
+Si 
+\begin_inset Formula $G$
 \end_inset
 
- con núcleo 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+ es un abierto de 
+\begin_inset Formula $\mathbb{R}^{n}$
 \end_inset
 
- es compacto, 
-\begin_inset Formula ${\cal C}([a,b])$
+ y 
+\begin_inset Formula $u,w\in L^{2}(G)$
 \end_inset
 
- es 
-\begin_inset Formula $K$
+, 
+\begin_inset Formula $w$
 \end_inset
 
--invariante y 
-\begin_inset Formula $K|_{{\cal C}([a,b])}:({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})\to({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$
+ es la 
+\series bold
+derivada generalizada 
+\begin_inset Formula $j$
 \end_inset
 
- es compacto.
-\begin_inset Note Note
-status open
+-ésima
+\series default
+ de 
+\begin_inset Formula $u$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+, 
+\begin_inset Formula $w=\partial_{j}u$
+\end_inset
+
+, si
+\begin_inset Formula 
+\[
+\forall v\in{\cal D}(G),\int_{G}u\partial_{j}v=-\int_{G}wv,
+\]
 
 \end_inset
 
+y para 
+\begin_inset Formula $\alpha\in\mathbb{N}^{n}$
+\end_inset
 
-\end_layout
+ llamamos 
+\begin_inset Formula $D^{\alpha}u\coloneqq\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}}u$
+\end_inset
 
-\begin_layout Section
-Teorema espectral
+.
+ 
 \end_layout
 
 \begin_layout Standard
-Como 
+Para 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $p\in[1,\infty)$
+\end_inset
+
+, llamamos 
 \series bold
-teorema
+espacio de Sobolev
 \series default
-, si 
-\begin_inset Formula $H$
+ a
+\begin_inset Formula 
+\[
+W^{k,p}(G)\coloneqq\{u\in L^{p}(G)\mid\forall\alpha\in\mathbb{N}^{n},(|\alpha|\leq k\implies\exists D^{\alpha}f\in L^{p}(G))\}.
+\]
+
 \end_inset
 
- es un 
-\begin_inset Formula $\mathbb{K}$
+Escribimos 
+\begin_inset Formula $W^{k}(G)\coloneqq W^{k,2}(G)$
 \end_inset
 
--espacio de Hilbert de dimensión finita y 
-\begin_inset Formula $T\in{\cal L}(H)$
+, y generalmente consideramos el espacio de Sobolev 
+\begin_inset Formula $W^{1}(G)$
 \end_inset
 
- es autoadjunto:
+.
 \end_layout
 
-\begin_layout Enumerate
+\begin_layout Standard
 Si 
-\begin_inset Formula $\lambda_{1},\dots,\lambda_{m}$
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
- son los distintos valores propios de 
-\begin_inset Formula $T$
+ es abierto, definimos la relación de equivalencia en 
+\begin_inset Formula $G\to\mathbb{R}$
 \end_inset
 
-, 
-\begin_inset Formula $H=\bigoplus_{k=1}^{m}\ker(T-\lambda_{k}I_{H})$
+ como 
+\begin_inset Formula $f\sim g\iff\{x\in G\mid f(x)\neq g(x)\}\text{ es de medida nula}$
+\end_inset
+
+, y 
+\begin_inset Formula $\langle\cdot,\cdot\rangle_{1,2}:W^{1}(G)/\sim\to\mathbb{R}$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+\langle\overline{u},\overline{v}\rangle_{1,2}\coloneqq\int_{G}\left(uv+\sum_{j}(\partial_{j}u)(\partial_{j}v)\right)
+\]
+
+\end_inset
+
+es un producto escalar en 
+\begin_inset Formula $W^{1}(G)/\sim$
+\end_inset
+
+ que lo convierte en un espacio de Hilbert.
+ Identificamos 
+\begin_inset Formula $W^{1}(G)$
+\end_inset
+
+ con 
+\begin_inset Formula $W^{1}(G)/\sim$
 \end_inset
 
 .
@@ -2982,122 +3908,209 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-Existe una base ortonormal 
-\begin_inset Formula $(e_{k})_{k}$
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula $H_{0}^{1}(G)$
 \end_inset
 
- de 
-\begin_inset Formula $H$
+ al espacio de Hilbert obtenido como la clausura de 
+\begin_inset Formula ${\cal D}(G)$
 \end_inset
 
- formada por vectores propios de 
-\begin_inset Formula $T$
+ en 
+\begin_inset Formula $W^{1}(G)$
+\end_inset
+
+, que en general es un subespacio propio de 
+\begin_inset Formula $W^{1}(G)$
+\end_inset
+
+ pero es igual a 
+\begin_inset Formula $W^{1}(G)$
+\end_inset
+
+ si 
+\begin_inset Formula $G=\mathbb{R}^{n}$
 \end_inset
 
 .
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $x\in X$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ es un abierto acotado no vacío y 
+\begin_inset Formula $u\in W^{1}(G)$
 \end_inset
 
 , 
-\begin_inset Formula $Tx=\sum_{k}\mu_{k}\langle x,e_{k}\rangle e_{k}$
+\series bold
+
+\begin_inset Formula $u$
 \end_inset
 
-, donde 
-\begin_inset Formula $\mu_{k}$
+ se anula en la frontera de 
+\begin_inset Formula $G$
 \end_inset
 
- es el valor propio asociado a 
-\begin_inset Formula $e_{k}$
+ en sentido generalizado
+\series default
+, 
+\begin_inset Formula $u=0$
+\end_inset
+
+ en 
+\begin_inset Formula $\partial G$
+\end_inset
+
+, si 
+\begin_inset Formula $u\in H_{0}^{1}(G)$
+\end_inset
+
+, y para 
+\begin_inset Formula $f,g\in W^{1}(G)$
+\end_inset
+
+, 
+\series bold
+
+\begin_inset Formula $f=g$
+\end_inset
+
+ en 
+\begin_inset Formula $\partial G$
+\end_inset
+
+ en sentido generalizado
+\series default
+ si 
+\begin_inset Formula $f-g\in H_{0}^{1}(G)$
 \end_inset
 
 .
 \end_layout
 
 \begin_layout Standard
-Si 
-\begin_inset Formula $T$
+
+\series bold
+Desigualdad de Poincaré-Friedrichs:
+\series default
+ Si 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
- es un operador compacto autoadjunto en el espacio de Hilbert 
-\begin_inset Formula $H$
+ es un abierto acotado no vacío, existe 
+\begin_inset Formula $C>0$
 \end_inset
 
-, 
-\begin_inset Formula $\Vert T\Vert$
+ tal que para 
+\begin_inset Formula $u\in H_{0}^{1}(G)$
 \end_inset
 
- o 
-\begin_inset Formula $-\Vert T\Vert$
+,
+\begin_inset Formula 
+\[
+C\int_{G}u^{2}\leq\int_{G}\sum_{j=1}^{n}(\partial_{j}u)^{2}.
+\]
+
 \end_inset
 
- es valor propio de 
-\begin_inset Formula $T$
+
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $R\coloneqq\prod_{i}[a_{i},b_{i}]$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+ con 
+\begin_inset Formula $G\subseteq R$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ y 
+\begin_inset Formula $u\in{\cal D}(G)$
+\end_inset
 
+, y vemos 
+\begin_inset Formula $u$
 \end_inset
 
+ como una función en 
+\begin_inset Formula $R$
+\end_inset
 
-\end_layout
+ que se anula fuera de 
+\begin_inset Formula $G$
+\end_inset
 
-\begin_layout Standard
-Todo operador normal compacto en un 
-\begin_inset Formula $\mathbb{C}$
+ y con valor indefinido en 
+\begin_inset Formula $\partial G$
 \end_inset
 
--espacio de Hilbert tiene algún valor propio.
-\begin_inset Note Note
-status open
+, para 
+\begin_inset Formula $x\in R$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+, por la desigualdad de Cauchy-Schwartz,
+\begin_inset Formula 
+\begin{align*}
+(u(x))^{2} & =\left(\int_{a_{n}}^{x_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)\dif t\right)^{2}\leq\left(\int_{a_{n}}^{x_{n}}\dif t\right)\left(\int_{a_{n}}^{x_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\right)\leq\\
+ & \leq(b_{n}-a_{n})\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t,
+\end{align*}
 
 \end_inset
 
+luego
+\begin_inset Formula 
+\begin{align*}
+\int_{G}u^{2} & =\int_{R}u^{2}\leq\int_{a_{1}}^{b_{1}}\cdots\int_{a_{n}}^{b_{n}}(b_{n}-a_{n})\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\dif x_{n}\cdots\dif x_{1}=\\
+ & =(b_{n}-a_{n})^{2}\int_{a_{1}}^{b_{1}}\cdots\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\dif x_{n-1}\cdots\dif x_{1}=\\
+ & =(b_{n}-a_{n})^{2}\int_{R}(\partial_{n}u)^{2}\dif x\leq(b_{n}-a_{n})^{2}\int_{R}\sum_{j}(\partial_{j}u)^{2}\dif x=(b_{n}-a_{n})^{2}\int_{G}\sum_{j}(\partial_{j}u)^{2}\dif x.
+\end{align*}
 
-\end_layout
+\end_inset
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $T\in{\cal L}(H)$
+Para 
+\begin_inset Formula $u\in H_{0}^{1}(G)$
 \end_inset
 
- es compacto en el 
-\begin_inset Formula $\mathbb{K}$
+,existe una sucesión 
+\begin_inset Formula $\{u_{m}\}_{m}\subseteq{\cal D}(G)$
 \end_inset
 
--espacio de Hilbert 
-\begin_inset Formula $H$
+ con 
+\begin_inset Formula $\lim_{m}\Vert u-u_{m}\Vert_{1,2}=0$
 \end_inset
 
- y 
-\begin_inset Formula $\lambda\in\mathbb{K}\setminus0$
+ y por tanto 
+\begin_inset Formula $\lim_{m}\Vert u-u_{m}\Vert_{2}=\lim_{m}\Vert\partial_{j}u-\partial_{j}u_{m}\Vert_{2}=0$
 \end_inset
 
-, 
-\begin_inset Formula $\ker(T-\lambda1_{H})$
+, y tomando límites y usando que la norma 
+\begin_inset Formula $\Vert\cdot\Vert_{2}\leq\Vert\cdot\Vert_{1,2}$
 \end_inset
 
- es de dimensión finita.
-\begin_inset Note Note
-status open
+ y por tanto es continua en 
+\begin_inset Formula $W^{1}(G)$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+,
+\begin_inset Formula 
+\[
+C\int_{G}u^{2}-\int_{G}\sum_{j}(\partial_{j}u)^{2}=C\Vert u\Vert_{2}^{2}-\sum_{j}\Vert\partial_{j}u\Vert_{2}^{2}=\lim_{m}\left(C\Vert u_{m}\Vert_{2}^{2}-\sum_{j}\Vert\partial_{j}u_{m}\Vert_{2}^{2}\right)\leq0.
+\]
 
 \end_inset
 
@@ -3105,126 +4118,255 @@ nproof
 \end_layout
 
 \begin_layout Standard
-Sean 
-\begin_inset Formula $X$
+
+\series bold
+Principio de Dirichlet:
+\series default
+ Sean 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
 \end_inset
 
- e 
-\begin_inset Formula $Y$
+ un abierto acotado no vacío, 
+\begin_inset Formula $f\in L^{2}(G)$
 \end_inset
 
- espacios de Banach y 
-\begin_inset Formula $T\in{\cal L}(X,Y)$
+ y 
+\begin_inset Formula $g\in W^{1}(G)$
 \end_inset
 
- compacto, 
-\begin_inset Formula $\sigma_{\text{p}}(T)$
+, 
+\begin_inset Formula $F:\{u\in W^{1}(G)\mid u-g\in H_{0}^{1}(G)\}\to\mathbb{R}$
 \end_inset
 
- es contable, contiene a 
-\begin_inset Formula $\sigma(T)\setminus\{0\}$
+ dada por 
+\begin_inset Formula 
+\[
+F(u)\coloneqq\frac{1}{2}\int_{G}\sum_{j=1}^{n}(\partial_{j}u)^{2}-\int_{G}fu
+\]
+
 \end_inset
 
- y, si es infinito, es una sucesión acotada con a lo sumo un punto de acumulació
-n, el 0, y si 
-\begin_inset Formula $T$
+alcanza su mínimo en un único punto, que es el único 
+\begin_inset Formula $u\in\text{Dom}f$
 \end_inset
 
- es normal el 0 es punto de acumulación.
-\begin_inset Note Note
-status open
+ tal que
+\begin_inset Formula 
+\[
+\forall v\in H_{0}^{1}(G),\int_{G}\sum_{j=1}^{n}(\partial_{j}u)(\partial_{j}v)=\int_{G}fv
+\]
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+\end_inset
 
+y la única solución en 
+\begin_inset Formula $\text{Dom}f$
 \end_inset
 
+ del problema de valores frontera para la ecuación de Poisson 
+\begin_inset Formula $-\nabla^{2}u=f$
+\end_inset
 
+.
 \end_layout
 
 \begin_layout Standard
 
 \series bold
-Teorema espectral para operadores compactos autoadjuntos:
+Demostración:
 \series default
- Sean 
-\begin_inset Formula $H$
+ Para 
+\begin_inset Formula $u,v\in W^{1}(G)$
 \end_inset
 
- un 
-\begin_inset Formula $\mathbb{K}$
+ definimos 
+\begin_inset Formula 
+\begin{align*}
+B(u,v) & \coloneqq\int_{G}\sum_{j}(\partial_{j}u)(\partial_{j}v), & b_{0}(v) & \coloneqq\int_{G}fv, & b(v) & \coloneqq b_{0}(v)-B(v,g).
+\end{align*}
+
 \end_inset
 
--espacio de Hilbert y 
-\begin_inset Formula $T\in{\cal L}(H)$
+
+\begin_inset Formula $B$
 \end_inset
 
- compacto normal:
-\end_layout
+ es bilineal y simétrica, y es acotada porque
+\begin_inset Formula 
+\[
+|B(u,v)|=\left|\sum_{j}\int_{G}(\partial_{j}u)(\partial_{j}v)\right|\leq\sum_{j}\left|\int_{G}(\partial_{j}u)(\partial_{j}v)\right|\leq\sum_{j}\Vert\partial_{j}u\Vert_{2}\Vert\partial_{j}v\Vert_{2}\leq n\Vert u\Vert_{1,2}\Vert v\Vert_{1,2}.
+\]
 
-\begin_layout Enumerate
-\begin_inset Formula $\sigma_{\text{p}}(T)\setminus\{0\}$
 \end_inset
 
- es contable.
-\begin_inset Note Note
-status open
+Por la desigualdad de Poincaré-Friedrichs, existe 
+\begin_inset Formula $C>0$
+\end_inset
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+ tal que, para todo 
+\begin_inset Formula $v\in H$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+C\int_{G}v^{2}\leq\int_{G}\sum_{j}(\partial_{j}v)^{2},
+\]
 
 \end_inset
 
+luego 
+\begin_inset Formula 
+\[
+C\Vert v\Vert_{1,2}^{2}=C\left(\int_{G}v^{2}+\sum_{j}(\partial_{j}v)^{2}\right)\leq(1+C)\int_{G}\sum_{j}(\partial_{j}v)^{2}=(1+C)B(v,v)
+\]
 
-\end_layout
+\end_inset
+
+y 
+\begin_inset Formula $B$
+\end_inset
+
+ es fuertemente positiva.
+ Además, 
+\begin_inset Formula $b_{0}$
+\end_inset
+
+ es lineal y es acotada por la desigualdad de Cauchy-Schwartz, y como además
+ 
+\begin_inset Formula $B$
+\end_inset
+
+ es bilineal y acotada, 
+\begin_inset Formula $b_{0}$
+\end_inset
+
+ es lineal acotada y se dan las condiciones del teorema principal de los
+ problemas variacionales cuadráticos.
+ Ahora bien, si 
+\begin_inset Formula $w\coloneqq u-g\in H_{0}^{1}(G)$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{multline*}
+\frac{1}{2}B(w,w)-b(w)=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}(u-g))^{2}-\int_{G}f(u-g)+\int_{G}\sum_{j}(\partial_{j}(u-g))(\partial_{j}(g))=\\
+=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}(u-g))(\partial_{j}(u+g))-\int_{G}f(u-g)=\\
+=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}u)^{2}-\int_{G}fu+\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}g)^{2}+\int_{G}fg,
+\end{multline*}
+
+\end_inset
+
+luego minimizar 
+\begin_inset Formula $F$
+\end_inset
+
+ equivale a minimizar 
+\begin_inset Formula $\frac{1}{2}B(w,w)-b(w)$
+\end_inset
+
+, y además
+\begin_inset Formula 
+\begin{multline*}
+B(w,v)=b(v)\iff B(u,v)-B(g,v)=b_{0}(v)-B(v,g)\iff B(u,v)=b_{0}(v)\iff\\
+\iff\int_{G}\sum_{j}(\partial_{j}u)(\partial_{j}v)=\int_{G}fv.
+\end{multline*}
+
+\end_inset
+
+Para la última parte, si 
+\begin_inset Formula $u_{0}$
+\end_inset
+
+ cumple esta última fórmula para todo 
+\begin_inset Formula $v\in H_{0}^{1}(G)$
+\end_inset
+
+, por integración por partes,
+\begin_inset Formula 
+\[
+0=\int_{G}\sum_{j}(\partial_{j}u_{0})(\partial_{j}v)-\int_{G}fv=-\int_{G}\sum_{j}(\partial_{j}\partial_{j}u_{0})v-\int_{G}fv=-\int_{G}(\nabla^{2}u_{0}+f)v,
+\]
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $P_{\lambda}\in{\cal L}(H)$
 \end_inset
 
- es la proyección ortogonal sobre 
-\begin_inset Formula $\ker(T-\lambda1_{H})$
+con lo que 
+\begin_inset Formula $(\nabla^{2}u_{0}+f)\bot H_{0}^{1}(G)$
+\end_inset
+
+ y, como 
+\begin_inset Formula ${\cal D}(G)\subseteq H_{0}^{1}(G)$
+\end_inset
+
+ es denso en 
+\begin_inset Formula $L^{2}(G)$
 \end_inset
 
 , 
-\begin_inset Formula $T=\sum_{\lambda\in\sigma_{\text{p}}(T)}\lambda P_{\lambda}$
+\begin_inset Formula $\nabla^{2}u_{0}+f=0$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
+\end_layout
 
-\begin_layout Plain Layout
-nproof
+\begin_layout Section
+Soluciones débiles
 \end_layout
 
+\begin_layout Standard
+Dados 
+\begin_inset Formula $k,n\in\mathbb{N}$
 \end_inset
 
+ y 
+\begin_inset Formula $a_{\alpha}\in\mathbb{K}^{n}$
+\end_inset
 
-\end_layout
+ para cada 
+\begin_inset Formula $\alpha\in\mathbb{N}^{n}$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $\overline{\text{Im}T}=\bigoplus_{\lambda\in\sigma_{\text{p}}(T)\setminus\{0\}}\ker(T-\lambda1_{H})$
+ con 
+\begin_inset Formula $|\alpha|<k$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
+, un 
+\series bold
+operador diferencial lineal de coeficientes constantes
+\series default
+ es uno de la forma
+\begin_inset Formula 
+\[
+L\coloneqq\sum_{|\alpha|\leq k}a_{\alpha}\left(\frac{\partial}{\partial x}\right)^{\alpha}\coloneqq\sum_{|\alpha|\leq k}a_{\alpha}\frac{\partial^{|\alpha|}}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}},
+\]
 
-\begin_layout Plain Layout
-nproof
-\end_layout
+\end_inset
+
+y su 
+\series bold
+operador adjunto
+\series default
+ es
+\begin_inset Formula 
+\[
+L^{*}\coloneqq\sum_{|\alpha|\leq k}(-1)^{|\alpha|}\overline{a_{\alpha}}\left(\frac{\partial}{\partial x}\right)^{\alpha}.
+\]
 
 \end_inset
 
+Si 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
+\end_inset
 
-\end_layout
+ es abierto, 
+\begin_inset Formula $\varphi,\psi\in L^{2}(G)$
+\end_inset
 
-\begin_layout Enumerate
-\begin_inset Formula $H=\ker T\oplus\overline{\text{Im}T}$
+ son de clase 
+\begin_inset Formula ${\cal C}^{k}$
+\end_inset
+
+ y una de las dos tiene soporte compacto, entonces 
+\begin_inset Formula $\langle L\psi,\varphi\rangle=\langle\psi,L^{*}\varphi\rangle$
 \end_inset
 
 .
@@ -3240,37 +4382,90 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-Existe una base ortonormal 
-\begin_inset Formula $(e_{n})_{n\in J}$
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $G$
 \end_inset
 
- de 
-\begin_inset Formula $\overline{\text{Im}T}$
+ es un abierto en 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $f,u\in L^{2}(G)$
+\end_inset
+
+ son de clase 
+\begin_inset Formula ${\cal C}^{k}$
 \end_inset
 
  y 
-\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{C}$
+\begin_inset Formula $Lu=f$
 \end_inset
 
- tales que, para 
-\begin_inset Formula $x\in H$
+, entonces 
+\begin_inset Formula $\langle f,\psi\rangle=\langle u,L^{*}\psi\rangle$
+\end_inset
+
+ para todo 
+\begin_inset Formula $\psi\in{\cal D}(G)$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $f\in L^{2}(G)$
+\end_inset
+
+, 
+\begin_inset Formula $u\in L^{2}(G)$
+\end_inset
+
+ es 
+\series bold
+solución débil
+\series default
+ de la ecuación en derivadas parciales 
+\begin_inset Formula $Lu=f$
+\end_inset
+
+ si para todo 
+\begin_inset Formula $\psi\in{\cal D}(G)$
+\end_inset
+
+ es 
+\begin_inset Formula $\langle f,\psi\rangle=\langle u,L^{*}\psi\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $L=\od{}{x}$
+\end_inset
+
+ y 
+\begin_inset Formula $u,f\in L^{2}((0,1))$
 \end_inset
 
 , 
-\begin_inset Formula $(\mu_{n}\langle x,e_{n}\rangle e_{n})_{n\in J}$
+\begin_inset Formula $Lu=f$
 \end_inset
 
- es sumable con suma 
-\begin_inset Formula $Tx$
+ en sentido débil si y sólo si existe 
+\begin_inset Formula $F:(0,1)\to\mathbb{R}$
 \end_inset
 
-, y entonces 
-\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\sigma_{\text{p}}(T)\setminus\{0\}$
+ absolutamente continua con 
+\begin_inset Formula $F=u$
 \end_inset
 
  y 
-\begin_inset Formula $\forall\lambda\in\sigma_{\text{p}}(T)\setminus\{0\},|\{n\in J\mid\mu_{n}=\lambda\}|=\dim\ker(T-\lambda1_{H})$
+\begin_inset Formula $F'=f$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $x\in(0,1)$
 \end_inset
 
 .
@@ -3286,20 +4481,25 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $P_{0}$
-\end_inset
+\begin_layout Standard
+La ecuación de ondas en una dimensión,
+\begin_inset Formula 
+\[
+\left\{ \begin{array}{rlrl}
+\frac{\partial^{2}u}{\partial x^{2}}-\frac{\partial^{2}u}{\partial t^{2}} & =0, & t & \in[0,+\infty),\\
+u(x,0) & \equiv f(x), & x & \in[0,\pi],\\
+\frac{\partial u}{\partial t}(x,0) & \equiv0,
+\end{array}\right.
+\]
 
- es la proyección ortogonal sobre 
-\begin_inset Formula $\ker T$
 \end_inset
 
-, 
-\begin_inset Formula $\forall x\in H,x=P_{0}x+\sum_{n\in J}\langle x,e_{n}\rangle e_{n}$
+siendo 
+\begin_inset Formula $f:[0,\pi]\to\mathbb{R}$
 \end_inset
 
-.
+ una función lineal a trozos, admite soluciones débiles que no son soluciones
+ ordinarias.
 \begin_inset Note Note
 status open
 
@@ -3313,356 +4513,462 @@ nproof
 \end_layout
 
 \begin_layout Standard
-Si 
-\begin_inset Formula $H$
-\end_inset
-
- es un 
-\begin_inset Formula $\mathbb{K}$
-\end_inset
 
--espacio de Hilbert, 
-\begin_inset Formula $T\in{\cal L}(H)$
+\series bold
+Teorema de Malgrange-Ehrenpreis:
+\series default
+ Sean 
+\begin_inset Formula $G$
 \end_inset
 
- es compacto autoadjunto si y sólo si hay una familia ortonormal contable
- 
-\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+ un abierto acotado de 
+\begin_inset Formula $\mathbb{R}^{n}$
 \end_inset
 
  y 
-\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\begin_inset Formula $L$
 \end_inset
 
- de modo que 
-\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+ un operador en derivadas parciales lineal con coeficientes constantes,
+ existe un operador lineal continuo 
+\begin_inset Formula $K:L^{2}(G)\to L^{2}(G)$
 \end_inset
 
- y 0 es el único punto de acumulación de 
-\begin_inset Formula $(\mu_{n})_{n}$
+ tal que para todo 
+\begin_inset Formula $f\in L^{2}(G)$
 \end_inset
 
-.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-nproof
-\end_layout
-
+, 
+\begin_inset Formula $u\coloneqq K(f)$
 \end_inset
 
+ es solución débil de 
+\begin_inset Formula $Lu=f$
+\end_inset
 
+.
 \end_layout
 
 \begin_layout Standard
 
 \series bold
-Teorema de alternativa de Fredholm:
+Demostración:
 \series default
- Sean 
-\begin_inset Formula $H$
+ Definimos 
+\begin_inset Formula $\langle\varphi,\psi\rangle_{L}\coloneqq\langle L^{*}\varphi,L^{*}\psi\rangle_{2}$
 \end_inset
 
- un 
-\begin_inset Formula $\mathbb{K}$
+, y para ver que es un producto escalar sobre 
+\begin_inset Formula ${\cal D}(G)$
 \end_inset
 
--espacio de Hilbert, 
-\begin_inset Formula $T\in{\cal L}(H)$
+ vemos que existe 
+\begin_inset Formula $C>0$
 \end_inset
 
- compacto autoadjunto, 
-\begin_inset Formula $(e_{n})_{n\in J}$
+ tal que, para 
+\begin_inset Formula $\psi\in{\cal D}(G)$
 \end_inset
 
- una base ortonormal de 
-\begin_inset Formula $\overline{\text{Im}T}$
+, 
+\begin_inset Formula $\Vert\psi\Vert_{2}\leq C\Vert L^{*}\psi\Vert_{2}$
 \end_inset
 
- de modo que 
-\begin_inset Formula $Tx\eqqcolon\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+.
+ Si 
+\begin_inset Formula $L^{*}=\frac{\partial}{\partial x_{1}}$
 \end_inset
 
- para ciertos 
-\begin_inset Formula $\mu_{n}\in\mathbb{K}$
+, llamando 
+\begin_inset Formula $\psi(x)\coloneqq0$
 \end_inset
 
- e 
-\begin_inset Formula $y\in H$
+ para 
+\begin_inset Formula $x\notin G$
 \end_inset
 
-:
-\end_layout
+, para 
+\begin_inset Formula $x\in G$
+\end_inset
 
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $\lambda\in\mathbb{K}\setminus\{\sigma_{\text{p}}(T)\cup\{0\})$
+, como 
+\begin_inset Formula $\text{sop}\psi\subseteq G$
 \end_inset
 
-, la ecuación 
-\begin_inset Formula $(\lambda1_{H}-T)x=y$
+ es compacto, sea 
+\begin_inset Formula $m\coloneqq\inf_{x\in G}x_{1}$
 \end_inset
 
- tiene como única solución
+,
 \begin_inset Formula 
-\[
-x=\frac{1}{\lambda}\left(y+\sum_{n\in J}\frac{\mu_{n}}{\lambda-\mu_{n}}\langle y,e_{n}\rangle e_{n}\right).
-\]
+\begin{align*}
+\psi(x)^{2} & =\left(\int_{m}^{x_{1}}\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\dif t\right)^{2}\leq\left(\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|\cdot1\dif t\right)\leq\\
+ & \leq\int_{m}^{x_{1}}\dif t\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}\dif t\leq d\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2},
+\end{align*}
 
 \end_inset
 
+donde 
+\begin_inset Formula $d$
+\end_inset
 
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Si existe solución 
-\begin_inset Formula $x\in H$
+ es el diámetro de 
+\begin_inset Formula $G$
 \end_inset
 
-, 
+, e integrando de nuevo,
 \begin_inset Formula 
-\[
-(\lambda1_{H}-T)x=y\iff\lambda x=Tx+y\iff x=\frac{1}{\lambda}\left(\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}+y\right),
-\]
+\begin{align*}
+\Vert\psi\Vert_{2}^{2} & =\int_{G}\psi(x)^{2}\dif x\leq d\int_{m}^{x_{1}}\int_{-\infty}^{x_{2}}\cdots\int_{-\infty}^{x_{n}}\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}\dif t\dif x_{n}\cdots\dif x_{1}\leq\\
+ & \leq d^{2}\int_{G}\left|\frac{\partial\psi}{\partial x_{1}}(x)\right|^{2}\dif x=d^{2}\Vert L^{*}\psi\Vert_{2}^{2}.
+\end{align*}
+
+\end_inset
+
+Si 
+\begin_inset Formula $L^{*}=\frac{\partial}{\partial x_{i}}$
+\end_inset
 
+ para otro 
+\begin_inset Formula $i$
 \end_inset
 
-pero entonces 
-\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\lambda}(\mu_{n}\langle x,e_{n}\rangle+\langle y,e_{n}\rangle)$
+, es análogo, y si 
+\begin_inset Formula $L^{*}=\left(\frac{\partial}{\partial x}\right)^{|\alpha|}$
+\end_inset
+
+, por inducción, 
+\begin_inset Formula $\Vert\psi\Vert_{2}\leq d^{|\alpha|}\Vert L^{*}\psi\Vert_{2}$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $L$
+\end_inset
+
+ arbitrario basta hacer combinaciones lineales.
+ Visto esto, sean 
+\begin_inset Formula $H_{0}\coloneqq({\cal D}(G),\langle\cdot,\cdot\rangle_{L})$
 \end_inset
 
  y 
-\begin_inset Formula $(\lambda-\mu_{n})\langle x,e_{n}\rangle=\langle y,e_{n}\rangle$
+\begin_inset Formula $H$
 \end_inset
 
-, y como 
-\begin_inset Formula $\lambda-\mu_{n}\neq0$
+ su compleción, 
+\begin_inset Formula $L^{*}:H_{0}\to L^{2}(G)$
 \end_inset
 
-, podemos sustituir 
-\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\lambda-\mu_{n}}\langle y,e_{n}\rangle$
+ es lineal y continuo y por tanto admite una extensión lineal y continua
+ 
+\begin_inset Formula $\hat{L}^{*}:H\to L^{2}(G)$
 \end_inset
 
- en lo anterior y queda la solución del enunciado.
- Queda ver que la serie converge, pero si 
-\begin_inset Formula $\sigma_{\text{p}}(T)$
+.
+ Sea ahora 
+\begin_inset Formula $f\in L^{2}(G)$
 \end_inset
 
- es infinito, 
-\begin_inset Formula $\{\mu_{n}\}_{n}\subseteq\sigma_{\text{p}}(T)$
+ y 
+\begin_inset Formula $l_{0}:H_{0}\to\mathbb{K}$
 \end_inset
 
- es acotado y por tanto lo es 
-\begin_inset Formula $\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|$
+ dada por 
+\begin_inset Formula $l_{0}(\psi)\coloneqq\langle\psi,f\rangle_{2}$
 \end_inset
 
- y
+,
 \begin_inset Formula 
 \[
-\sum_{n\in J}\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|^{2}|\langle y,e_{n}\rangle|^{2}\leq\sup_{n\in J}\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|^{2}\sum_{n\in J}|\langle y,e_{n}\rangle|^{2}<\infty.
+|l_{0}(\psi)|=|\langle\psi,f\rangle_{2}|\leq\Vert\psi\Vert_{2}\Vert f\Vert_{2}\leq C\Vert f\Vert_{2}\Vert L^{*}\psi\Vert_{2},
 \]
 
 \end_inset
 
+donde 
+\begin_inset Formula $C$
+\end_inset
 
-\end_layout
+ es tal que 
+\begin_inset Formula $\Vert\psi\Vert_{2}\leq C\Vert L^{*}\psi\Vert_{2}$
+\end_inset
 
-\end_deeper
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $\lambda\in\sigma_{\text{p}}(T)\setminus\{0\}$
+ para todo 
+\begin_inset Formula $C$
 \end_inset
 
-, la ecuación 
-\begin_inset Formula $(\lambda1_{H}-T)x=y$
+, de modo que 
+\begin_inset Formula $l_{0}$
 \end_inset
 
- tiene solución si y sólo si 
-\begin_inset Formula $y\bot\ker(\lambda1_{H}-T)$
+ es lineal continua por la cota 
+\begin_inset Formula $C\Vert f\Vert_{2}$
 \end_inset
 
-, en cuyo caso las soluciones son
-\begin_inset Formula 
-\begin{align*}
-x & =\frac{1}{\lambda}\left(y+\sum_{\begin{subarray}{c}
-n\in J\\
-\mu_{n}\neq\lambda
-\end{subarray}}\frac{\mu_{n}}{\lambda-\mu_{n}}\langle y,e_{n}\rangle e_{n}\right)+z, & z & \in\ker(\lambda1_{H}-T).
-\end{align*}
+ y se puede extender a una forma lineal y continua 
+\begin_inset Formula $l:H\to\mathbb{K}$
+\end_inset
 
+ con 
+\begin_inset Formula $\Vert l\Vert\leq C\Vert f\Vert_{2}$
 \end_inset
 
+.
+ Por el teorema de Riesz, existe un único 
+\begin_inset Formula $\hat{u}\in H$
+\end_inset
 
-\end_layout
+ con 
+\begin_inset Formula $l(h)\equiv\langle h,\hat{u}\rangle_{L}$
+\end_inset
 
-\begin_deeper
-\begin_layout Standard
-Si la ecuación tiene solución 
-\begin_inset Formula $x$
+ para 
+\begin_inset Formula $h\in H$
 \end_inset
 
-, entonces 
-\begin_inset Formula $y=(\lambda1_{H}-T)x\in\text{Im}(\lambda1_{H}-T)\subseteq\overline{\text{Im}(\lambda1_{H}-T)}=\ker((\lambda1_{H}-T)^{*})^{\bot}=\ker(\lambda1_{H}-T)^{\bot}$
+ y además 
+\begin_inset Formula $\Vert\hat{u}\Vert_{H}=\Vert l\Vert_{H}$
 \end_inset
 
- por ser 
-\begin_inset Formula $1_{H}$
+, y tomando 
+\begin_inset Formula $u\coloneqq\hat{L}^{*}\hat{u}$
+\end_inset
+
+, 
+\begin_inset Formula $l(h)=\langle\hat{L}^{*}h,\hat{L}^{*}\hat{u}\rangle=\langle\hat{L}^{*}h,u\rangle_{2}$
+\end_inset
+
+, pero para 
+\begin_inset Formula $\psi\in{\cal D}(G)$
+\end_inset
+
+, 
+\begin_inset Formula $l(\psi)=\langle\psi,f\rangle_{2}$
 \end_inset
 
  y 
-\begin_inset Formula $T$
+\begin_inset Formula $\hat{L}^{*}(\psi)=L^{*}\psi$
 \end_inset
 
- autoadjuntos, y claramente dos soluciones difieren en un vector de 
-\begin_inset Formula $\ker(\lambda1_{H}-T)$
+, con lo que 
+\begin_inset Formula $\langle L^{*}\psi,u\rangle_{2}=l(\psi)=\langle\psi,f\rangle_{2}$
+\end_inset
+
+, y basta llamar 
+\begin_inset Formula $K(f)\coloneqq u$
 \end_inset
 
 .
- Queda ver que, si 
-\begin_inset Formula $y\in\ker(\lambda1_{H}-T)^{\bot}$
+ Para la continuidad de 
+\begin_inset Formula $K$
 \end_inset
 
-, la 
-\begin_inset Formula $x$
+,
+\begin_inset Formula 
+\[
+\Vert K(f)\Vert_{2}=\Vert u\Vert_{2}=\Vert\hat{L}^{*}\hat{u}\Vert_{2}=\Vert\hat{u}\Vert_{H}=\Vert l\Vert_{H}=\sup_{\Vert\psi\Vert_{H}=\Vert L^{*}\psi\Vert_{2}=1}|l(\psi)|\leq C\Vert f\Vert_{2}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Método de Galerkin
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $M_{1}\subseteq M_{2}\subseteq\dots\subseteq M_{n}\subseteq\dots$
 \end_inset
 
- del enunciado es solución, para lo cual hacemos la misma sustitución que
- al principio del primer apartado pero, cuando 
-\begin_inset Formula $\lambda=\mu_{n}$
+ una sucesión de subespacios cerrados de un espacio de Hilbert 
+\begin_inset Formula $H$
 \end_inset
 
-, en su lugar vemos que 
-\begin_inset Formula $(\lambda-\mu_{n})\langle x,e_{n}\rangle=\langle y,e_{n}\rangle$
+ con unión densa en 
+\begin_inset Formula $H$
 \end_inset
 
- y por tanto 
-\begin_inset Formula $\langle y,e_{n}\rangle=0$
+, 
+\begin_inset Formula $a:H\times H\to\mathbb{R}$
 \end_inset
 
-, por lo que excluimos dicho factor de la serie, la cual converge por el
- mismo motivo que en el primer apartado y resulta en la solución del enunciado.
-\end_layout
+ bilineal, simétrica, continua y fuertemente positiva, 
+\begin_inset Formula $b:H\to\mathbb{R}$
+\end_inset
+
+ lineal continua,
+\begin_inset Formula 
+\[
+J(x)\coloneqq\frac{1}{2}a(x,x)-b(x)
+\]
 
-\end_deeper
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $y=0$
+\end_inset
+
+para 
+\begin_inset Formula $x\in H$
 \end_inset
 
 , 
-\begin_inset Formula $Tx=y$
+\begin_inset Formula $u\in H$
 \end_inset
 
- tiene solución si y sólo si 
-\begin_inset Formula $y\bot\ker T$
+ con 
+\begin_inset Formula $J(u)$
 \end_inset
 
- y 
-\begin_inset Formula $\sum_{n\in J}\left|\frac{\langle y,e_{n}\rangle}{\mu_{n}}\right|^{2}<\infty$
+ mínimo y, para 
+\begin_inset Formula $n\in\mathbb{N}$
 \end_inset
 
-, en cuyo caso las soluciones son
-\begin_inset Formula 
-\begin{align*}
-x & =\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+z, & z & \in\ker T.
-\end{align*}
+, 
+\begin_inset Formula $u_{n}\in M_{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $J(u_{n})$
+\end_inset
+
+ mínimo, de modo que 
+\begin_inset Formula $a(x,u_{n})=b(x)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $x\in M_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $a(x,u)=b(x)$
+\end_inset
 
+ para todo 
+\begin_inset Formula $x\in H$
 \end_inset
 
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Teorema de Galerkin-Ritz:
+\series default
+ 
+\begin_inset Formula $\lim_{n}u_{n}=u$
+\end_inset
 
+.
 \end_layout
 
 \begin_deeper
 \begin_layout Standard
-Si la ecuación tiene solución 
-\begin_inset Formula $x$
+Para 
+\begin_inset Formula $x\in M_{n}$
 \end_inset
 
 , 
-\begin_inset Formula $y\in\text{Im}T\subseteq(\ker T)^{\bot}$
+\begin_inset Formula $a(x,u_{n})=b(x)$
 \end_inset
 
- y
-\begin_inset Formula 
-\[
-\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}=Tx=y=\sum_{n\in J}\langle y,e_{n}\rangle e_{n},
-\]
+, y para 
+\begin_inset Formula $x\in H$
+\end_inset
 
+, 
+\begin_inset Formula $a(x,u)=f(x)$
 \end_inset
 
-con lo que 
-\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\mu_{n}}\langle y,e_{n}\rangle$
+, luego 
+\begin_inset Formula $a(x,u-u_{n})=b(x)-b(x)=0$
 \end_inset
 
- para cada 
-\begin_inset Formula $n$
+ para 
+\begin_inset Formula $x\in M_{n}$
 \end_inset
 
- y por tanto 
-\begin_inset Formula $\sum_{n\in J}\left|\frac{\langle y,e_{n}\rangle}{\mu_{n}}\right|^{2}=\Vert x\Vert^{2}<\infty$
+.
+ Pero 
+\begin_inset Formula $a$
 \end_inset
 
-, y como 
-\begin_inset Formula $(e_{n})_{n}$
+ es un producto escalar equivalente al de 
+\begin_inset Formula $H$
 \end_inset
 
- es base de 
-\begin_inset Formula $\overline{\text{Im}T}$
+, luego 
+\begin_inset Formula $u-u_{n}\bot M_{n}$
 \end_inset
 
-, 
-\begin_inset Formula $x\in\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+\overline{\text{Im}T}^{\bot}$
+ y, si 
+\begin_inset Formula $P_{n}:H\to M_{n}$
 \end_inset
 
- con 
-\begin_inset Formula $\overline{\text{Im}T}^{\bot}=\ker T$
+ es la proyección ortogonal, 
+\begin_inset Formula $P_{n}(u)=u_{n}$
 \end_inset
 
 .
- Finalmente, si esta condición se cumple, 
-\begin_inset Formula $y\in\overline{\text{Im}T}$
+ Por el teorema de la proyección, 
+\begin_inset Formula $\Vert u-u_{n}\Vert=\Vert u-P_{n}(u)\Vert=d(u,M_{n})$
 \end_inset
 
-, la serie del enunciado converge y
-\begin_inset Formula 
-\[
-T\left(\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+z\right)=\sum_{n\in J}\langle y,e_{n}\rangle e_{n}+0=y.
-\]
+, pero por la densidad es 
+\begin_inset Formula $d(u,\bigcup_{n}M_{n})=0$
+\end_inset
 
+, y para 
+\begin_inset Formula $\varepsilon>0$
 \end_inset
 
+ existen 
+\begin_inset Formula $n_{0}\in\mathbb{N}$
+\end_inset
 
-\end_layout
+ e 
+\begin_inset Formula $y\in M_{n_{0}}$
+\end_inset
 
-\end_deeper
-\begin_layout Standard
-Sea 
-\begin_inset Formula $A$
+ con 
+\begin_inset Formula $\Vert u-y\Vert<\varepsilon$
 \end_inset
 
- un operador en un espacio de Hilbert 
-\begin_inset Formula $H$
+, y como la sucesión es creciente, para 
+\begin_inset Formula $n\geq n_{0}$
 \end_inset
 
-:
+, 
+\begin_inset Formula $\Vert u-u_{n}\Vert=d(u,M_{n})\leq d(u,M_{n_{0}})\leq\Vert u-y\Vert<\varepsilon$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\lim_{n}u_{n}=u$
+\end_inset
+
+.
 \end_layout
 
+\end_deeper
 \begin_layout Enumerate
-\begin_inset Formula $A$
+Dados 
+\begin_inset Formula $c,d>0$
 \end_inset
 
- es una isometría si y sólo si 
-\begin_inset Formula $A^{*}$
+ con 
+\begin_inset Formula $a(x,y)\leq d\Vert x\Vert\Vert y\Vert$
 \end_inset
 
- es inverso por la izquierda de 
-\begin_inset Formula $A$
+ y 
+\begin_inset Formula $c\Vert x\Vert^{2}\leq a(x,x)$
 \end_inset
 
-, si y sólo si 
-\begin_inset Formula $\forall x,y\in H,\langle Ax,Ay\rangle=\langle x,y\rangle$
+ para todo 
+\begin_inset Formula $x,y\in H$
+\end_inset
+
+, 
+\begin_inset Formula $c\Vert u\Vert\leq\Vert b\Vert$
 \end_inset
 
 .
@@ -3679,27 +4985,118 @@ nproof
 \end_layout
 
 \begin_layout Enumerate
-\begin_inset Formula $A$
+
+\series bold
+Razón de convergencia:
+\series default
+ 
+\begin_inset Formula $\Vert u-u_{n}\Vert\leq\frac{d}{c}d(u,M_{n})$
 \end_inset
 
- es un isomorfismo isométrico, si y sólo si es una isometría suprayectiva,
- si y sólo si 
-\begin_inset Formula $A^{*}$
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Estimación del error:
+\series default
+ Si 
+\begin_inset Formula $\beta\leq J(x)$
 \end_inset
 
- es inverso de 
-\begin_inset Formula $A$
+ para todo 
+\begin_inset Formula $x\in H$
 \end_inset
 
-, y entonces decimos que 
-\begin_inset Formula $A$
+, para 
+\begin_inset Formula $n\in\mathbb{N}$
 \end_inset
 
  es 
+\begin_inset Formula $\frac{c}{2}\Vert u-u_{n}\Vert^{2}\leq J(u_{n})-\beta$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El 
 \series bold
-unitario
+método de Galerkin
 \series default
+ para resolver un problema de esta forma consiste en tomar en el teorema
+ anterior los 
+\begin_inset Formula $M_{n}$
+\end_inset
+
+ de dimensión finita y resolver los sistemas de ecuaciones lineales resultantes,
+ con matriz de coeficientes simétrica y definida positiva de tamaño 
+\begin_inset Formula $\dim M_{n}$
+\end_inset
+
 .
+ Tomando adecuadamente las bases de los 
+\begin_inset Formula $M_{n}$
+\end_inset
+
+ se puede conseguir que las matrices tengan muchas entradas nulas.
+\end_layout
+
+\begin_layout Section
+Bases hilbertianas
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $(H_{i})_{i\in I}$
+\end_inset
+
+ una familia de 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert, 
+\begin_inset Formula $H_{0}\coloneqq\prod_{i\in I}H_{i}$
+\end_inset
+
+ y 
+\begin_inset Formula $\langle\cdot,\cdot\rangle:H_{0}\times H_{0}\to[0,+\infty]$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+\langle x,y\rangle\coloneqq\sum_{i\in I}\langle x_{i},y_{i}\rangle_{H_{i}},
+\]
+
+\end_inset
+
+llamamos 
+\series bold
+suma directa hilbertiana
+\series default
+ o 
+\series bold
+suma 
+\begin_inset Formula $\ell^{2}$
+\end_inset
+
+
+\series default
+ de 
+\begin_inset Formula $\{H_{i}\}_{i\in I}$
+\end_inset
+
+ al espacio de Hilbert 
+\begin_inset Formula 
+\[
+\bigoplus_{i\in I}H_{i}\coloneqq\ell^{2}((H_{i})_{i\in I})\coloneqq(\{x\in H_{0}\mid\langle x,x\rangle<\infty\},\langle\cdot,\cdot\rangle).
+\]
+
+\end_inset
+
+
 \begin_inset Note Note
 status open
 
@@ -3713,28 +5110,84 @@ nproof
 \end_layout
 
 \begin_layout Standard
-Sean 
+Cada 
+\begin_inset Formula $H_{i}$
+\end_inset
+
+ es isométricamente isomorfo al subespacio de 
 \begin_inset Formula $H$
 \end_inset
 
- un 
+ de los vectores con todas las coordenadas nulas salvo la 
+\begin_inset Formula $i$
+\end_inset
+
+, los 
+\begin_inset Formula $H_{i}$
+\end_inset
+
+ son mutuamente ortogonales en 
+\begin_inset Formula $H$
+\end_inset
+
+, 
+\begin_inset Formula $H$
+\end_inset
+
+ es la clausura lineal cerrada de los 
+\begin_inset Formula $H_{i}$
+\end_inset
+
+ y cada 
+\begin_inset Formula $x\in H$
+\end_inset
+
+ se puede expresar de forma única como 
+\begin_inset Formula $\sum_{i\in I}x_{i}$
+\end_inset
+
+ con cada 
+\begin_inset Formula $x_{i}\in H_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
 \begin_inset Formula $\mathbb{K}$
 \end_inset
 
 -espacio de Hilbert y 
-\begin_inset Formula $S,T\in{\cal L}(H)$
+\begin_inset Formula $(H_{i})_{i\in I}$
+\end_inset
+
+ es una familia de subespacios cerrados de 
+\begin_inset Formula $H$
 \end_inset
 
- compactos autoadjuntos, 
-\begin_inset Formula $\forall\lambda\in\mathbb{K},\dim\ker(T-\lambda1_{H})=\dim\ker(S-\lambda1_{H})$
+ mutuamente ortogonales con 
+\begin_inset Formula $H=\overline{\text{span}\{H_{i}\}_{i\in I}}$
 \end_inset
 
- si y sólo si existe 
-\begin_inset Formula $U\in{\cal L}(H)$
+, entonces 
+\begin_inset Formula $H$
+\end_inset
+
+ es isométricamente isomorfo a 
+\begin_inset Formula $\bigoplus_{i\in I}H_{i}$
+\end_inset
+
+, e identificamos 
+\begin_inset Formula $H$
 \end_inset
 
- unitario con 
-\begin_inset Formula $U^{*}SU=T$
+ con 
+\begin_inset Formula $\bigoplus_{i\in I}H_{i}$
 \end_inset
 
 .
@@ -3751,80 +5204,268 @@ nproof
 \end_layout
 
 \begin_layout Standard
-\begin_inset Formula $S,T\in{\cal L}(H)$
+
+\series bold
+Desigualdad de Bessel:
+\series default
+ Sean 
+\begin_inset Formula $H$
 \end_inset
 
- en el 
-\begin_inset Formula $\mathbb{K}$
+ un espacio prehilbertiano y 
+\begin_inset Formula $\{e_{i}\}_{i\in I}\subseteq H$
+\end_inset
+
+ una familia ortonormal, para 
+\begin_inset Formula $x\in H$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}\leq\Vert x\Vert^{2}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Para un conjunto 
+\begin_inset Formula $I$
+\end_inset
+
+ arbitrario, llamamos 
+\begin_inset Formula $\ell^{2}(I)\coloneqq\bigoplus_{i\in I}\mathbb{K}$
 \end_inset
 
--espacio de Hilbert 
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de la base hilbertiana:
+\series default
+ Sean 
 \begin_inset Formula $H$
 \end_inset
 
- son 
+ un espacio de Hilbert y 
+\begin_inset Formula $\{e_{i}\}_{i\in I}\subseteq H$
+\end_inset
+
+ una familia ortonormal, 
+\begin_inset Formula $\{e_{i}\}_{i\in I}$
+\end_inset
+
+ es ortonormal maximal (por inclusión) si y sólo si 
+\begin_inset Formula $\forall x\in H,(\forall i\in I,\langle x,e_{i}\rangle=0\implies x=0)$
+\end_inset
+
+, si y sólo si es un conjunto total, si y sólo si 
+\begin_inset Formula $\hat{}:H\to\ell^{2}(I)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\hat{x}\coloneqq(\langle x,e_{i}\rangle)_{i\in I}$
+\end_inset
+
+ es inyectiva, si y sólo si todo 
+\begin_inset Formula $x\in H$
+\end_inset
+
+ admite un 
 \series bold
-simultáneamente diagonalizables
+desarrollo de Fourier
 \series default
- si existe una familia ortonormal 
-\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+ 
+\begin_inset Formula $x=\sum_{i\in I}\langle x,e_{i}\rangle e_{i}$
 \end_inset
 
- y 
-\begin_inset Formula $\{\alpha_{n}\}_{n\in J},\{\beta_{n}\}_{n\in J}\subseteq\mathbb{K}$
+, si y sólo si 
+\begin_inset Formula $\forall x,y\in H,\langle x,y\rangle=\sum_{i\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{i}\rangle}$
 \end_inset
 
- tal que
-\begin_inset Formula 
-\[
-\forall x\in H,\left(Sx=\sum_{n\in J}\alpha_{n}\langle x,e_{n}\rangle e_{n}\land Tx=\sum_{n\in J}\beta_{n}\langle x,e_{n}\rangle e_{n}\right).
-\]
+, si y sólo si todo 
+\begin_inset Formula $x\in H$
+\end_inset
 
+ cumple la 
+\series bold
+identidad de Parseval
+\series default
+, 
+\begin_inset Formula $\Vert x\Vert^{2}=\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}$
 \end_inset
 
-Si 
-\begin_inset Formula $S$
+, y entonces decimos que 
+\begin_inset Formula $(e_{i})_{i\in I}$
 \end_inset
 
- y 
-\begin_inset Formula $T$
+ es una 
+\series bold
+base hilbertiana
+\series default
+ de 
+\begin_inset Formula $H$
+\end_inset
+
+ o un 
+\series bold
+sistema ortonormal completo
+\series default
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ Entonces 
+\begin_inset Formula $x\bot\{e_{i}\}_{i\in I}$
+\end_inset
+
+, por lo que si 
+\begin_inset Formula $x\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $\{e_{i}\}_{i\in I}\cup\{x\}$
+\end_inset
+
+ sería ortogonal.
+\begin_inset Formula $\#$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\iff3]$
+\end_inset
+
+ Sabemos que un 
+\begin_inset Formula $S\subseteq H$
 \end_inset
 
- son compactos y autoadjuntos esto equivale a que 
-\begin_inset Formula $ST=TS$
+ es total si y sólo si 
+\begin_inset Formula $S^{\bot}=0$
 \end_inset
 
 .
-\begin_inset Note Note
-status open
+\end_layout
 
-\begin_layout Plain Layout
-nproof
+\begin_layout Description
+\begin_inset Formula $2\iff4]$
+\end_inset
+
+ Por ser 
+\begin_inset Formula $\hat{}$
+\end_inset
+
+ lineal.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $4\implies5]$
+\end_inset
+
+ 
+\begin_inset Formula $\widehat{\sum_{i}\langle x,e_{i}\rangle e_{i}}=\sum_{i}\langle x,e_{i}\rangle\hat{e}_{i}=\sum_{i}\langle x,e_{i}\rangle e_{i}=\hat{x}$
+\end_inset
+
+, y por inyectividad 
+\begin_inset Formula $x=\sum_{i\in I}\langle x,e_{i}\rangle e_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $5\implies6]$
+\end_inset
+
+ 
+\begin_inset Formula $\langle x,y\rangle=\sum_{i,j\in I}\langle\langle x,e_{i}\rangle e_{i},\langle y,e_{j}\rangle e_{j}\rangle=\sum_{i,j\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{j}\rangle}\langle e_{i},e_{j}\rangle=\sum_{i\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{j}\rangle}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $6\implies7]$
+\end_inset
+
+ Basta tomar 
+\begin_inset Formula $x=y$
+\end_inset
+
+.
 \end_layout
 
+\begin_layout Description
+\begin_inset Formula $7\implies1]$
+\end_inset
+
+ Si fuera 
+\begin_inset Formula $\{e_{i}\}_{i}\subsetneq M\subseteq H$
+\end_inset
+
+ con 
+\begin_inset Formula $M$
+\end_inset
+
+ ortonormal, para 
+\begin_inset Formula $x\in M\setminus\{e_{i}\}_{i}$
 \end_inset
 
+, 
+\begin_inset Formula $1=\Vert x\Vert^{2}=\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}=0\#$
+\end_inset
 
+.
 \end_layout
 
 \begin_layout Standard
 
 \series bold
-Teorema espectral para operadores compactos normales:
+Primer teorema de Riesz-Fischer:
 \series default
  Si 
 \begin_inset Formula $H$
 \end_inset
 
- es un 
-\begin_inset Formula $\mathbb{C}$
+ es un espacio prehilbertiano con una familia ortonormal 
+\begin_inset Formula $\{e_{i}\}_{i\in I}$
 \end_inset
 
--espacio de Hilbert y 
-\begin_inset Formula $T\in{\cal L}(H)$
+ y 
+\begin_inset Formula $\hat{}:H\to\mathbb{K}^{I}$
+\end_inset
+
+ viene dada por 
+\begin_inset Formula $\hat{x}\coloneqq(\langle x,e_{i}\rangle)_{i\in I}$
+\end_inset
+
+, 
+\begin_inset Formula $\hat{}$
+\end_inset
+
+ es lineal y continua con imagen contenida en 
+\begin_inset Formula $\ell^{2}(I)$
+\end_inset
+
+ e igual a 
+\begin_inset Formula $\ell^{2}(I)$
+\end_inset
+
+ si 
+\begin_inset Formula $H$
 \end_inset
 
- compacto normal, ocurre lo mismo que en el anterior teorema espectral.
+ es de Hilbert.
 \begin_inset Note Note
 status open
 
@@ -3842,24 +5483,17 @@ Si
 \begin_inset Formula $H$
 \end_inset
 
- es un 
-\begin_inset Formula $\mathbb{C}$
-\end_inset
-
--espacio de Hilbert, 
-\begin_inset Formula $T\in{\cal L}(H)$
-\end_inset
-
- es compacto normal si y sólo si hay una familia ortonormal contable 
-\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+ es un espacio de Hilbert, todo espacio ortonormal de vectores en 
+\begin_inset Formula $H$
 \end_inset
 
- y 
-\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{C}$
+ se puede completar a una base hilbertiana de 
+\begin_inset Formula $H$
 \end_inset
 
- con 0 como único punto de acumulación de modo que 
-\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+, y en particular todo espacio de Hilbert posee una base hilbertiana y es
+ isométricamente isomorfo a un 
+\begin_inset Formula $\ell^{2}(I)$
 \end_inset
 
 .
@@ -3876,28 +5510,16 @@ nproof
 \end_layout
 
 \begin_layout Standard
-Un operador entre 
-\begin_inset Formula $\mathbb{K}$
-\end_inset
-
--espacios de Hilbert 
-\begin_inset Formula $T\in{\cal L}(G,H)$
-\end_inset
-
- es compacto si y sólo si hay una familia contable 
-\begin_inset Formula $\{\nu_{n}\}_{n\in J}\subseteq\mathbb{R}^{+}$
-\end_inset
-
- con 0 como punto de acumulación, 
-\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq G$
+Los espacios de Hilbert 
+\begin_inset Formula $\ell^{2}(I)$
 \end_inset
 
  y 
-\begin_inset Formula $\{f_{n}\}_{n\in J}\subseteq H$
+\begin_inset Formula $\ell^{2}(J)$
 \end_inset
 
- tales que 
-\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\nu_{n}\langle x,e_{n}\rangle f_{n}$
+ son topológicamente isomorfos si y sólo si 
+\begin_inset Formula $|I|=|J|$
 \end_inset
 
 .
@@ -3913,135 +5535,156 @@ nproof
 
 \end_layout
 
-\begin_layout Section
-Ecuaciones integrales de Fredholm
-\end_layout
-
 \begin_layout Standard
-Una 
+Llamamos 
 \series bold
-ecuación integral de Fredholm
+dimensión hilbertiana
 \series default
- es una de la forma
-\begin_inset Formula 
-\[
-x(t)-\mu\int_{a}^{b}k(t,s)x(s)\dif s=g(t),
-\]
-
+ de un espacio de Hilbert al cardinal de cualquier base hilbertiana.
+ 
+\series bold
+Segundo teorema de Riesz-Fischer:
+\series default
+ Si 
+\begin_inset Formula $H$
 \end_inset
 
-donde 
-\begin_inset Formula $x,g\in L^{2}([a,b])$
+ es de dimensión infinita, 
+\begin_inset Formula $\dim H=\aleph_{0}\coloneqq|\mathbb{N}|$
 \end_inset
 
-, 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+ si y sólo si 
+\begin_inset Formula $H\cong\ell^{2}$
 \end_inset
 
- y la incógnita es 
-\begin_inset Formula $x$
+, si y sólo si 
+\begin_inset Formula $H$
 \end_inset
 
-.
- 
+ es separable.
 \end_layout
 
-\begin_layout Standard
-Un núcleo 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\begin_layout Description
+\begin_inset Formula $1\iff2]$
 \end_inset
 
- es 
-\series bold
-simétrico
-\series default
- si 
-\begin_inset Formula $k(t,s)=\overline{k(s,t)}$
+ Por lo anterior.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies3]$
 \end_inset
 
- para casi todo 
-\begin_inset Formula $s,t\in[a,b]$
+ Visto.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies2]$
 \end_inset
 
-.
- 
-\series bold
-Teorema de alternativa de Fredholm:
-\series default
- Sean 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+ Dado 
+\begin_inset Formula $\{x_{n}\}_{n\in\mathbb{N}}\subseteq H$
 \end_inset
 
- un núcleo simétrico, 
-\begin_inset Formula $K$
+ denso, como 
+\begin_inset Formula $H$
 \end_inset
 
- el operador integral asociado y 
-\begin_inset Formula $g\in L^{2}([a,b])$
+ es de dimensión infinita, existe una subsucesión 
+\begin_inset Formula $(x_{n_{k}})_{k}$
 \end_inset
 
-, si 
-\begin_inset Formula $Kx=\sum_{n\in J}\mu_{j}\langle x,e_{n}\rangle e_{n}$
+ linealmente independiente de 
+\begin_inset Formula $(x_{n})_{n}$
 \end_inset
 
- para cierta base hilbertiana contable 
-\begin_inset Formula $(e_{n})_{n\in J}$
+ con 
+\begin_inset Formula $\text{span}\{x_{n}\}_{n}=\text{span}\{x_{n_{k}}\}_{k}$
 \end_inset
 
- de 
-\begin_inset Formula $\overline{\text{Im}K}$
+, luego 
+\begin_inset Formula $\overline{\text{span}\{x_{n_{k}}\}_{k}}=H$
 \end_inset
 
-, ciertos 
-\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+ y el proceso de ortonormalización de Gram-Schmidt nos da una base hilbertiana
+ numerable de 
+\begin_inset Formula $H$
 \end_inset
 
- y todo 
-\begin_inset Formula $x\in X$
+.
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $Z\leq_{\mathbb{K}}\ell^{2}$
 \end_inset
 
-, considerando la ecuación integral de Fredholm de arriba, 
-\begin_inset Formula $x-Kx=g$
+ es cerrado de dimensión infinita, 
+\begin_inset Formula $Z\cong\ell^{2}$
 \end_inset
 
-:
+.
 \end_layout
 
-\begin_layout Enumerate
+\begin_layout Section
+Aproximaciones por polinomios
+\end_layout
+
+\begin_layout Standard
 Si 
-\begin_inset Formula $\mu=0$
+\begin_inset Formula $I\subseteq\mathbb{R}$
+\end_inset
+
+ es un intervalo cerrado, llamamos 
+\begin_inset Formula ${\cal C}(I)$
+\end_inset
+
+ al conjunto de funciones 
+\begin_inset Formula $I\to\mathbb{R}$
 \end_inset
 
-, la ecuación tiene como única solución 
-\begin_inset Formula $x=g$
+ continuas en el interior de 
+\begin_inset Formula $I$
 \end_inset
 
 .
 \end_layout
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $\frac{1}{\mu}\notin\{\mu_{n}\}_{n}$
+\begin_layout Standard
+
+\series bold
+Teorema de Korovkin:
+\series default
+ Sean 
+\begin_inset Formula $p_{0},p_{1},p_{2}:[a,b]\subseteq\mathbb{R}\to\mathbb{R}$
 \end_inset
 
-, la ecuación tiene como única solución
-\begin_inset Formula 
-\[
-x(t)=g(t)+\mu\left(\sum_{n}\frac{\mu_{n}}{1-\mu\mu_{n}}\left(\int_{a}^{b}g\overline{e_{n}}\right)e_{n}(t)\right),
-\]
+ dadas por 
+\begin_inset Formula $p_{k}(t)\coloneqq t^{k}$
+\end_inset
 
+ y 
+\begin_inset Formula $(P_{n}:{\cal C}([a,b])\to{\cal C}([a,b]))_{n}$
 \end_inset
 
-y existe 
-\begin_inset Formula $\alpha>0$
+ una sucesión de funciones lineales positivas (
+\begin_inset Formula $\forall f\in{\cal C}([a,b]),(f\geq0\implies P_{n}(f)\geq0)$
 \end_inset
 
- que depende solo de 
-\begin_inset Formula $k$
+) con 
+\begin_inset Formula $\lim_{n}\Vert P_{n}(p_{k})-p_{k}\Vert_{\infty}=0$
 \end_inset
 
- tal que 
-\begin_inset Formula $\Vert x\Vert_{2}\leq\alpha\Vert g\Vert_{2}$
+ para 
+\begin_inset Formula $k\in\{0,1,2\}$
+\end_inset
+
+, entonces, para 
+\begin_inset Formula $f\in{\cal C}([a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $\lim_{n}\Vert P_{n}(f)-f\Vert_{\infty}=0$
 \end_inset
 
 .
@@ -4057,27 +5700,26 @@ nproof
 
 \end_layout
 
-\begin_layout Enumerate
-Si existe 
-\begin_inset Formula $n\in J$
-\end_inset
+\begin_layout Standard
 
- con 
-\begin_inset Formula $\mu_{n}=\frac{1}{\mu}$
+\series bold
+Teorema de Weierstrass:
+\series default
+ El conjunto de polinomios en una variable es denso 
+\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$
 \end_inset
 
-, la ecuación tiene solución si y sólo si 
-\begin_inset Formula $g\bot\ker(\frac{1_{L^{2}([a,b])}}{\mu}-K)$
+, y en particular 
+\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$
 \end_inset
 
-, y entonces las soluciones son
-\begin_inset Formula 
-\begin{align*}
-x(t) & =g(t)+\mu\sum_{\begin{subarray}{c}
-n\in J\\
-\mu_{n}\neq\frac{1}{\mu}
-\end{subarray}}\frac{\mu_{n}}{1-\mu\mu_{n}}\left(\int g\overline{e_{n}}\right)e_{j}+u, & u & \in\ker(\tfrac{1_{L^{2}([a,b])}}{\mu}-K).
-\end{align*}
+ es separable.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
 
 \end_inset
 
@@ -4085,167 +5727,272 @@ n\in J\\
 \end_layout
 
 \begin_layout Standard
-La convergencia de las series es de media cuadrática, pero en ciertos casos
- puede ser uniforme.
+Así, para 
+\begin_inset Formula $f\in{\cal C}([a,b])$
+\end_inset
+
+, se puede encontrar una sucesión de polinomios que converja uniformemente
+ a 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Hacerlo con polinomios de interpolación por nodos prefijados no es una
+ buena estrategia ya que para toda secuencia de nodos de interpolación en
+ 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+, existe 
+\begin_inset Formula $f\in{\cal C}([a,b])$
+\end_inset
+
+ para la que los polinomios de interpolación en dichos nodos no converge
+ uniformemente a 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
 \end_inset
 
- es un núcleo simétrico con
-\begin_inset Formula 
-\[
-\sup_{t\in[a,b]}\int_{a}^{b}|k(t,s)|^{2}\dif s<\infty,
-\]
+ Si se hace con nodos equidistantes se da el fenómeno de Runge.
+\end_layout
+
+\begin_layout Standard
 
+\series bold
+Teorema de Čebyšev:
+\series default
+ Para 
+\begin_inset Formula $f\in{\cal C}([a,b])$
 \end_inset
 
+ y 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
 
-\begin_inset Formula $K$
+, si 
+\begin_inset Formula $K_{n}\subseteq\mathbb{K}[X]$
 \end_inset
 
- es el operador integral asociado y hay una base hilbertiana 
-\begin_inset Formula $(e_{n})_{n\in J}$
+ es el conjunto de polinomio de grado máximo 
+\begin_inset Formula $n$
 \end_inset
 
- de 
-\begin_inset Formula $\overline{\text{Im}K}$
+, 
+\begin_inset Formula $p:K_{n}\mapsto\Vert f-p\Vert_{\infty}$
 \end_inset
 
- y 
-\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+ tiene un único mínimo 
+\begin_inset Formula $p_{n}$
 \end_inset
 
- y tales que 
-\begin_inset Formula $Kx=\sum_{n}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+, y 
+\begin_inset Formula $(p_{n})_{n}$
 \end_inset
 
-:
+ converge uniformemente a 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
-\begin_layout Enumerate
+\end_inset
 
+
+\end_layout
+
+\begin_layout Standard
+Un 
 \series bold
-Teorema de Hilbert-Schmidt:
+polinomio trigonométrico real
 \series default
- Para 
-\begin_inset Formula $x\in L^{2}([a,b])$
+ es una función 
+\begin_inset Formula $p:\mathbb{R}\to\mathbb{R}$
 \end_inset
 
-,
+ de la forma
 \begin_inset Formula 
 \[
-\int_{a}^{b}k(t,s)x(s)\dif s=\sum_{n\in J}\mu_{n}\left(\int_{a}^{b}x\overline{e_{n}}\right)e_{n}(t)
+p(x)\coloneqq\sum_{n=0}^{m}(a_{n}\cos(nx)+b_{n}\sin(nx))
 \]
 
 \end_inset
 
-para casi todo 
-\begin_inset Formula $t\in[a,b]$
+para ciertos 
+\begin_inset Formula $a_{n},b_{n}\in\mathbb{R}$
 \end_inset
 
-, y si 
-\begin_inset Formula $J$
+.
+ 
+\series bold
+Teorema de Weierstrass:
+\series default
+ Si 
+\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{R}$
 \end_inset
 
- es numerable la serie converge absoluta y uniformemente en 
-\begin_inset Formula $[a,b]$
+ es continua con 
+\begin_inset Formula $f(-\pi)=f(\pi)$
+\end_inset
+
+, para cada 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ existe un polinomio trigonométrico real 
+\begin_inset Formula $p$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert f-p\Vert_{\infty}<\varepsilon$
 \end_inset
 
 .
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
-\begin_deeper
 \begin_layout Standard
-Para la primera parte basta tomar en el teorema anterior un 
-\begin_inset Formula $\mu\neq0$
+Para 
+\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{C}$
 \end_inset
 
- tal que 
-\begin_inset Formula $\frac{1}{\mu}$
+ integrable y 
+\begin_inset Formula $r\in\mathbb{Z}$
 \end_inset
 
- no sea valor propio y despejar.
- Para la segunda podemos suponer 
-\begin_inset Formula $J=(\mathbb{N},\geq)$
+, llamamos 
+\series bold
+
+\begin_inset Formula $r$
 \end_inset
 
-, y queremos ver que
+-ésimo coeficiente de Fourier
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ a
 \begin_inset Formula 
 \[
-\sum_{n}\left|\mu_{n}\left(\int_{a}^{b}x\overline{e_{n}}\right)e_{n}(t)\right|=\sum_{n}|\mu_{n}\langle x,e_{n}\rangle e_{n}(t)|
+\hat{f}(r)\coloneqq\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)\text{e}^{-\text{i}rt}\dif t,
 \]
 
 \end_inset
 
-es uniformemente de Cauchy en 
-\begin_inset Formula $[a,b]$
+y 
+\series bold
+serie de Fourier
+\series default
+ de 
+\begin_inset Formula $f$
 \end_inset
 
-.
- Por la desigualdad de Cauchy-Schwartz,
+ a la serie formal
 \begin_inset Formula 
 \[
-\sum_{n=p}^{q}|\mu_{n}e_{n}(t)||\langle x,e_{n}\rangle|\leq\sqrt{\sum_{n=p}^{q}|\mu_{n}e_{n}(t)|^{2}\sum_{n=p}^{q}|\langle x,e_{n}\rangle|^{2}},
+\sum_{r\in\mathbb{Z}}\hat{f}(r)\text{e}^{-\text{i}rt}.
 \]
 
 \end_inset
 
-pero para 
-\begin_inset Formula $n\in J$
+
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{R}$
 \end_inset
 
- y 
-\begin_inset Formula $t\in[a,b]$
+ integrable y 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
 \end_inset
 
-,
+, llamando
 \begin_inset Formula 
-\[
-\mu_{n}e_{n}(t)=K(e_{n})(t)=\int_{a}^{b}k(t,s)e_{k}(s)\dif s=\langle e_{k},\overline{k_{t}}\rangle,
-\]
+\begin{align*}
+a_{0} & \coloneqq\frac{1}{2\pi}\int_{-\pi}^{\pi}f, & a_{n} & \coloneqq\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(nt)\dif t, & b_{n} & \coloneqq\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\sin(nt)\dif t,
+\end{align*}
 
 \end_inset
 
-donde 
-\begin_inset Formula $k_{t}(s)\coloneqq k(t,s)$
+la 
+\series bold
+serie de Fourier real
+\series default
+ de 
+\begin_inset Formula $f$
 \end_inset
 
-, luego
+ es
 \begin_inset Formula 
 \[
-\sqrt{\sum_{n=p}^{q}|\mu_{n}e_{n}(t)|^{2}}=\sqrt{\sum_{n=p}^{q}|\langle e_{n},\overline{k_{t}}\rangle|^{2}}\leq\Vert k_{t}\Vert_{2}\leq\sup_{t\in[a,b]}\Vert k_{t}\Vert_{2}<\infty,
+\sum_{n=0}^{\infty}a_{n}\cos(nt)+\sum_{n=1}^{\infty}b_{n}\sin(nt).
 \]
 
 \end_inset
 
-con lo que esto está acotado superiormente por un valor independiente de
- 
-\begin_inset Formula $t$
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sean 
+\begin_inset Formula $([-\pi,\pi],\Sigma,\mu)$
 \end_inset
 
- y el resultado sale de que 
-\begin_inset Formula $|\langle x,e_{n}\rangle|^{2}$
+ es el espacio de medida usual en 
+\begin_inset Formula $[-\pi,\pi]$
 \end_inset
 
- tampoco depende de 
-\begin_inset Formula $t$
+, 
+\begin_inset Formula $M_{\mathbb{R}}\coloneqq L_{\mathbb{R}}^{2}([-\pi,\pi],\Sigma,\frac{\mu}{\pi})$
 \end_inset
 
  y 
-\begin_inset Formula $\lim_{p,q}\sum_{n=p}^{q}|\langle x,e_{n}\rangle|^{2}=0$
+\begin_inset Formula $M_{\mathbb{C}}\coloneqq L_{\mathbb{C}}^{2}([-\pi,\pi],\Sigma,\frac{\mu}{2\pi})$
 \end_inset
 
-.
+:
 \end_layout
 
-\end_deeper
 \begin_layout Enumerate
-Las series del teorema de alternativa de Fredholm convergen absoluta y uniformem
-ente en 
-\begin_inset Formula $[a,b]$
+El 
+\series bold
+sistema trigonométrico
+\series default
+ 
+\begin_inset Formula $(\text{e}^{\text{i}rt})_{r\in\mathbb{Z}}$
+\end_inset
+
+ es una base hilbertiana de 
+\begin_inset Formula $M_{\mathbb{C}}$
 \end_inset
 
 .
@@ -4261,44 +6008,67 @@ nproof
 
 \end_layout
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $k\in{\cal C}([a,b]\times[a,b])$
+\begin_layout Enumerate
+\begin_inset Formula $(\cos(nt))_{n\in\mathbb{N}}\star(\sin(nt))_{n\in\mathbb{N}^{*}}$
 \end_inset
 
- es un núcleo simétrico, existen una familia ortonormal contable 
-\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq({\cal C}([a,b]),\Vert\cdot\Vert_{2})$
+ es una base hilbertiana de 
+\begin_inset Formula $M_{\mathbb{R}}$
 \end_inset
 
- y 
-\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
- tales que, si 
-\begin_inset Formula $K$
+
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $f\in M_{\mathbb{C}}$
 \end_inset
 
- es el operador integral asociado a 
-\begin_inset Formula $k$
+, 
+\begin_inset Formula $f$
 \end_inset
 
- y 
-\begin_inset Formula $f\in{\cal C}([a,b])$
+ coincide con su serie de Fourier en 
+\begin_inset Formula $\Vert\cdot\Vert_{2}$
 \end_inset
 
-,
-\begin_inset Formula 
-\[
-Kf(t)=\sum_{n\in J}\mu_{n}\left(\int_{a}^{b}f\overline{e_{n}}\right)e_{n}(t)
-\]
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $f\in M_{\mathbb{R}}$
+\end_inset
 
+, 
+\begin_inset Formula $f$
 \end_inset
 
-para todo 
-\begin_inset Formula $t\in[a,b]$
+ coincide con su serie de Fourier real en 
+\begin_inset Formula $\Vert\cdot\Vert_{2}$
 \end_inset
 
- y la convergencia de la serie es absoluta y uniforme.
+.
 \begin_inset Note Note
 status open
 
@@ -4311,142 +6081,263 @@ nproof
 
 \end_layout
 
-\begin_layout Section
-Problemas de Sturm-Liouville
+\begin_layout Enumerate
+\begin_inset Formula ${\cal F}:M_{\mathbb{C}}\to\ell^{2}(\mathbb{Z})$
+\end_inset
+
+ que asigna a cada función su familia de coeficientes de Fourier 
+\begin_inset Formula $(\hat{f}(n))_{n\in\mathbb{Z}}$
+\end_inset
+
+ es un isomorfismo de espacios de Hilbert.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
 Un 
 \series bold
-problema regular de Sturm-Liouville
+peso
 \series default
+ en un intervalo cerrado 
+\begin_inset Formula $I\subseteq\mathbb{R}$
+\end_inset
+
+ es una 
+\begin_inset Formula $p\in{\cal C}(I)$
+\end_inset
+
+ estrictamente positiva tal que
+\begin_inset Formula 
+\[
+\forall n\in\mathbb{N},\int_{I}|t|^{n}p(t)\dif t<\infty.
+\]
+
+\end_inset
 
-\begin_inset Foot
+Entonces 
+\begin_inset Formula $\langle\cdot,\cdot\rangle:{\cal C}(I)\times{\cal C}(I)\to[-\infty,+\infty]$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+\langle f,g\rangle\coloneqq\int_{I}f\overline{g}p
+\]
+
+\end_inset
+
+es un producto escalar en 
+\begin_inset Formula $H_{p}\coloneqq\{f\in{\cal C}(I)\mid\langle f,f\rangle<\infty\}$
+\end_inset
+
+.
+\begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-La forma general del problema tiene como ecuación 
-\begin_inset Formula $\od{}{x}(p\dot{x})+qx+\lambda\sigma x+y=0$
-\end_inset
+nproof
+\end_layout
 
- con 
-\begin_inset Formula $p$
 \end_inset
 
- y 
-\begin_inset Formula $\sigma$
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+sucesión de polinomios ortonormales
+\series default
+ asociada a 
+\begin_inset Formula $\langle\cdot,\cdot\rangle$
 \end_inset
 
- continuas y estrictamente positivas.
- Aquí tomamos 
+ o al peso 
 \begin_inset Formula $p$
 \end_inset
 
- y 
-\begin_inset Formula $q$
+ en 
+\begin_inset Formula $I$
 \end_inset
 
- constantes en 1.
-\end_layout
+ a una sucesión 
+\begin_inset Formula $\{P_{n}\}_{n\in\mathbb{N}}\subseteq H_{p}$
+\end_inset
 
+ de polinomios con 
+\begin_inset Formula $\text{span}\{1,t,\dots,t^{n}\}=\text{span}\{P_{0},P_{1},\dots,P_{n}\}$
 \end_inset
 
- es uno de la forma
-\begin_inset Formula 
-\begin{align*}
--\ddot{x}+qx-\lambda x & =y, & \alpha x(a)+\beta\dot{x}(a) & =0, & \gamma x(b)+\delta\dot{x}(b) & =0,
-\end{align*}
+ para cada 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
 
+, y entonces, para 
+\begin_inset Formula $n\in\mathbb{N}$
 \end_inset
 
-donde 
-\begin_inset Formula $q\in{\cal C}([a,b],\mathbb{R})$
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $P_{n}$
 \end_inset
 
-, 
-\begin_inset Formula $y\in{\cal C}([a,b],\mathbb{C})$
+ es un polinomio de grado 
+\begin_inset Formula $n$
 \end_inset
 
-, 
-\begin_inset Formula $\lambda\in\mathbb{C}$
+ con coeficientes reales.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
-, 
-\begin_inset Formula $\alpha,\beta,\gamma,\delta\in\mathbb{R}$
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $P_{n}$
 \end_inset
 
- con 
-\begin_inset Formula $|\alpha|+|\beta|,|\gamma|+|\delta|\neq0$
+ es ortogonal en 
+\begin_inset Formula $H_{p}$
 \end_inset
 
- y la incógnita 
-\begin_inset Formula $x\in{\cal C}^{2}([a,b],\mathbb{C})$
+ al subespacio de polinomios de grado menor que 
+\begin_inset Formula $n$
 \end_inset
 
 .
- Su 
-\series bold
-operador de Sturm-Liouville
-\series default
- asociado es 
-\begin_inset Formula $S\in{\cal L}(D_{S},{\cal C}([a,b],\mathbb{C}))$
-\end_inset
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
 
- dado por 
-\begin_inset Formula $S(x)\coloneqq-\ddot{x}+qx$
 \end_inset
 
-, donde
-\begin_inset Formula 
-\[
-D_{S}\coloneqq\{x\in{\cal C}^{2}([a,b],\mathbb{C})\mid\alpha x(a)+\beta\dot{x}(a)=\gamma x(b)+\delta\dot{x}(b)=0\},
-\]
 
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $P_{n}$
+\end_inset
+
+ tiene 
+\begin_inset Formula $n$
 \end_inset
 
-y entonces el problema anterior es 
-\begin_inset Formula $(S-\mu1_{D_{S}})x=y$
+ raíces distintas en 
+\begin_inset Formula $(a,b)$
 \end_inset
 
 .
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
-Para 
-\begin_inset Formula $q\in{\cal C}([a,b],\mathbb{R})$
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Polinomios de Legendre:
+\series default
+ 
+\begin_inset Formula $I=[-1,1]$
 \end_inset
 
- e 
-\begin_inset Formula $y_{0},y_{1}\in\mathbb{R}$
+, 
+\begin_inset Formula $p(t)=1$
 \end_inset
 
-, el problema de Cauchy
-\begin_inset Formula 
-\begin{align*}
--\ddot{x}+qx & =0, & x(a) & =y_{0}, & \dot{x}(a) & =y_{1}
-\end{align*}
+, 
+\begin_inset Formula $P_{n}(t)=\frac{\sqrt{\frac{2n+1}{2}}}{2^{n}n!}\od[n]{(t^{2}-1)^{n}}{t}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
 
 \end_inset
 
-tiene una única solución real, y para 
-\begin_inset Formula $\alpha,\beta\in\mathbb{R}$
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Polinomios de Laguerre:
+\series default
+ 
+\begin_inset Formula $I=[0,\infty)$
 \end_inset
 
- con 
-\begin_inset Formula $|\alpha|+|\beta|\neq0$
+, 
+\begin_inset Formula $p(t)=\text{e}^{-t}$
 \end_inset
 
-, si 
-\begin_inset Formula $(y_{0},y_{1})\in\mathbb{R}^{2}$
+, 
+\begin_inset Formula $P_{n}(t)=\frac{\text{e}^{t}}{n!}\od[n]{\text{e}^{-t}t^{n}}{t}$
 \end_inset
 
- recorre la recta 
-\begin_inset Formula $\alpha y_{0}+\beta y_{1}=0$
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
-, la correspondiente solución del problema recorre una recta (subespacio
- de dimensión 1) de 
-\begin_inset Formula ${\cal C}^{2}([a,b])$
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Polinomios de Hermite:
+\series default
+ 
+\begin_inset Formula $I=\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $p(t)=\text{e}^{-t^{2}}$
+\end_inset
+
+, 
+\begin_inset Formula $P_{n}(t)=\frac{\text{e}^{t^{2}}}{\sqrt[4]{\pi}\sqrt{2^{n}n!}}\od[n]{\text{e}^{-t^{2}}}{t}$
 \end_inset
 
 .
@@ -4462,105 +6353,172 @@ nproof
 
 \end_layout
 
-\begin_layout Standard
-El 
+\begin_layout Enumerate
+
 \series bold
-determinante wronskiano
+Polinomios de Čebyšev:
 \series default
- de 
-\begin_inset Formula $x_{1},\dots,x_{n}\in{\cal C}^{n-1}([a,b],\mathbb{K})$
+ 
+\begin_inset Formula $I=[-1,1]$
 \end_inset
 
- es 
-\begin_inset Formula $W(x_{1},\dots,x_{n}):[a,b]\to\mathbb{K}$
+, 
+\begin_inset Formula $p(t)=\frac{1}{\sqrt{1-t^{2}}}$
 \end_inset
 
- dada por 
-\begin_inset Formula $t\mapsto\det(x_{j}^{(i)}(t))_{0\leq i<n}^{1\leq j\leq n}$
+, 
+\begin_inset Formula $P_{n}(t)=\cos(n\arccos t)$
+\end_inset
+
+, siendo 
+\begin_inset Formula $\arccos:[-1,1]\to[0,\pi]$
 \end_inset
 
 .
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
-Si 
-\begin_inset Formula $S:D_{S}\to{\cal C}([a,b],\mathbb{C})$
+Una sucesión de polinomios ortonormales asociada a un peso 
+\begin_inset Formula $p$
 \end_inset
 
- es un operador de Sturm-Liouville asociado al problema con parámetros 
-\begin_inset Formula $q,y,\lambda,\alpha,\beta,\gamma,\delta$
+ en un intervalo compacto es total en 
+\begin_inset Formula $H_{p}$
 \end_inset
 
-, existen 
-\begin_inset Formula $u,v\in{\cal C}([a,b],\mathbb{R})$
+, y en particular los polinomios de Legendre forman una base hilbertiana
+ en 
+\begin_inset Formula $L^{2}([-1,1]).$
 \end_inset
 
- con 
-\begin_inset Formula $-\ddot{u}+qu=0$
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
-, 
-\begin_inset Formula $\alpha x(a)+\beta\dot{x}(a)=0$
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $p$
 \end_inset
 
-, 
-\begin_inset Formula $-\ddot{v}+qv=0$
+ es un peso en 
+\begin_inset Formula $[a,b]$
 \end_inset
 
  y 
-\begin_inset Formula $\gamma x(b)+\delta\dot{x}(b)=0$
+\begin_inset Formula $a\leq t_{1}<\dots<t_{n}\leq b$
 \end_inset
 
-, y entonces 
-\begin_inset Formula $W(u,v)(t)$
-\end_inset
+, se tiene una 
+\series bold
+fórmula de cuadratura gaussiana
+\series default
+, 
+\begin_inset Formula 
+\[
+\int_{a}^{b}fp\approx\sum_{k=1}^{n}A_{k}f(t_{k})
+\]
 
- es constante en 
-\begin_inset Formula $t$
 \end_inset
 
- y, si 
-\begin_inset Formula $S$
+para ciertos 
+\begin_inset Formula $A_{1},\dots,A_{n}\in\mathbb{R}$
 \end_inset
 
- es inyectivo, 
-\begin_inset Formula $W(u,v)(t)\neq0$
+, y se alcanza la igualdad si 
+\begin_inset Formula $f$
 \end_inset
 
- y 
-\begin_inset Formula $u$
+ es un polinomio de grado menor que 
+\begin_inset Formula $n$
 \end_inset
 
- y 
-\begin_inset Formula $v$
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
- son linealmente independientes, y llamamos 
+
+\end_layout
+
+\begin_layout Standard
+
 \series bold
-función de Green
+Teorema de Gauss:
 \series default
- asociada a 
-\begin_inset Formula $S$
+ Dados un peso 
+\begin_inset Formula $p$
 \end_inset
 
- al núcleo simétrico 
-\begin_inset Formula $k\in{\cal C}([a,b]\times[a,b])$
+ en 
+\begin_inset Formula $[a,b]$
 \end_inset
 
- dado por
+ con una sucesión de polinomios ortonormales 
+\begin_inset Formula $(P_{n})_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $a<t_{1}<\dots<t_{n}<b$
+\end_inset
+
+ y 
+\begin_inset Formula $A_{1},\dots,A_{n}\in\mathbb{R}$
+\end_inset
+
+, si
 \begin_inset Formula 
 \[
-k(t,s)\coloneqq-\frac{u(\min\{t,s\})v(\max\{t,s\})}{W(u,v)(a)},
+\int_{a}^{b}fp=\sum_{k=1}^{n}A_{k}f(t_{k})
 \]
 
 \end_inset
 
-que no depende de 
-\begin_inset Formula $u$
+para todo polinomio 
+\begin_inset Formula $f$
 \end_inset
 
- y 
-\begin_inset Formula $v$
+ de grado menor que 
+\begin_inset Formula $n$
+\end_inset
+
+, esta fórmula se para polinomios de grado menor que 
+\begin_inset Formula $2n$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $t_{1},\dots,t_{n}$
+\end_inset
+
+ son los ceros de 
+\begin_inset Formula $P_{n}$
 \end_inset
 
 .
@@ -4577,39 +6535,52 @@ nproof
 \end_layout
 
 \begin_layout Standard
-Si 
-\begin_inset Formula $S:D_{S}\to{\cal C}([a,b])$
+
+\series bold
+Teorema de Stieltjes:
+\series default
+ Sean 
+\begin_inset Formula $p$
 \end_inset
 
- es un operador de Sturm-Liouville inyectivo con función de Green 
-\begin_inset Formula $k$
+ un peso en 
+\begin_inset Formula $[a,b]$
 \end_inset
 
-, llamamos 
-\series bold
-operador de Green
-\series default
- asociado a 
-\begin_inset Formula $S$
+ con una sucesión de polinomios ortonormales 
+\begin_inset Formula $(P_{n})_{n}$
 \end_inset
 
- al operador integral 
-\begin_inset Formula $G:L^{2}([a,b])\to L^{2}([a,b])$
+ y, para 
+\begin_inset Formula $n\in\mathbb{N}$
 \end_inset
 
- asociado al núcleo 
-\begin_inset Formula $k$
+, 
+\begin_inset Formula $t_{n1}<\dots<t_{nn}$
 \end_inset
 
-, y entonces 
-\begin_inset Formula $G|_{{\cal C}([a,b])}$
+ los ceros de 
+\begin_inset Formula $P_{n}$
 \end_inset
 
- es el inverso de 
-\begin_inset Formula $S$
+ y 
+\begin_inset Formula $A_{n1},\dots,A_{nn}\in\mathbb{R}$
 \end_inset
 
-.
+ los correspondientes coeficientes en la fórmula de cuadratura gaussiana,
+ para 
+\begin_inset Formula $f\in{\cal C}([a,b])$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\int_{a}^{b}fp=\lim_{n}\sum_{k=1}^{n}A_{nk}f(t_{nk}).
+\]
+
+\end_inset
+
+
 \begin_inset Note Note
 status open
 
@@ -4622,125 +6593,221 @@ nproof
 
 \end_layout
 
+\begin_layout Section
+El espacio de Bergman
+\end_layout
+
 \begin_layout Standard
-Así, 
-\begin_inset Formula $(S-\mu1_{D_{S}})x=y$
+Llamamos 
+\begin_inset Formula $D(a,r)\coloneqq B(a,r)\subseteq\mathbb{C}$
 \end_inset
 
- tiene solución única 
-\begin_inset Formula $x\in D_{S}$
+.
+ Si 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
 \end_inset
 
- si y sólo si 
-\begin_inset Formula $(1_{{\cal C}([a,b])}-\mu G)x=Gy$
+ es abierto, 
+\begin_inset Formula ${\cal H}(\Omega)$
+\end_inset
+
+ es el conjunto de las funciones holomorfas en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, y para 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{D(a,r)}\subseteq\Omega$
+\end_inset
+
+, la serie 
+\begin_inset Formula $\sum_{n\in\mathbb{N}}a_{n}(z-a)^{n}$
 \end_inset
 
- tiene solución única 
-\begin_inset Formula $x\in{\cal C}([a,b])$
+ con 
+\begin_inset Formula $z\in D(a,r)$
+\end_inset
+
+ converge uniformemente a 
+\begin_inset Formula $f$
+\end_inset
+
+ en compactos de 
+\begin_inset Formula $D(a,r)$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $a_{n}\in\mathbb{C}$
 \end_inset
 
 .
 \end_layout
 
 \begin_layout Standard
-Como 
+Si 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
+\end_inset
+
+ es abierto, llamamos 
+\begin_inset Formula ${\cal T}_{\text{K}}$
+\end_inset
+
+ a la topología en 
+\begin_inset Formula ${\cal H}(\Omega)$
+\end_inset
+
+ de convergencia uniforme sobre compactos, y 
 \series bold
-teorema
+espacio de Bergman
 \series default
-, si 
-\begin_inset Formula $S:D_{S}\to{\cal C}([a,b],\mathbb{C})$
+ en el abierto 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
 \end_inset
 
- es el operador de Sturm-Liouville asociado al problema con parámetros 
-\begin_inset Formula $q,y,\lambda,\alpha,\beta,\gamma,\delta$
+ a
+\begin_inset Formula 
+\[
+A^{2}(\Omega)\coloneqq\left\{ f\in{\cal H}(\Omega)\;\middle|\;\int_{\Omega}|f|^{2}<\infty\right\} ,
+\]
+
 \end_inset
 
-, existe una sucesión 
-\begin_inset Formula $(\nu_{n})_{n}$
+un subespacio cerrado y separable de 
+\begin_inset Formula $L^{2}(\Omega)$
 \end_inset
 
- de reales distintos con 
-\begin_inset Formula $\sum_{n}\frac{1}{\nu_{n}^{2}}<\infty$
+ que es pues un espacio de Hilbert numerable con 
+\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$
 \end_inset
 
- y una base hilbertiana numerable 
-\begin_inset Formula $(u_{n})_{n}$
+, y en el que la topología inducida por 
+\begin_inset Formula $L^{2}(\Omega)$
 \end_inset
 
- de 
-\begin_inset Formula $L^{2}([a,b])$
+ es más fina que la inducida por 
+\begin_inset Formula ${\cal T}_{\text{K}}$
 \end_inset
 
- tales que:
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
-\begin_layout Enumerate
-\begin_inset Formula $\forall n\in\mathbb{N},Su_{n}=\nu_{n}u_{n}$
 \end_inset
 
-.
+
 \end_layout
 
-\begin_layout Enumerate
-\begin_inset Formula 
-\[
-\forall x\in D_{S},\forall t\in[a,b],x(t)=\sum_{n}\left(\int_{a}^{b}xu_{n}\right)u_{n}(t),
-\]
+\begin_layout Standard
+Si 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
+\end_inset
 
+ es abierto, 
+\begin_inset Formula $(\omega_{n})_{n}$
 \end_inset
 
-donde la serie converge absoluta y uniformemente para 
-\begin_inset Formula $t\in[a,b]$
+ es base hilbertiana de 
+\begin_inset Formula $A^{2}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in A^{2}(\Omega)$
+\end_inset
+
+, el desarrollo en serie de Fourier de 
+\begin_inset Formula $f$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{n}\langle f,\omega_{n}\rangle\omega_{n}$
+\end_inset
+
+, converge uniformemente a 
+\begin_inset Formula $f$
+\end_inset
+
+ en compactos de 
+\begin_inset Formula $\Omega$
 \end_inset
 
 .
- 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
 \end_layout
 
-\begin_layout Enumerate
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Si 
-\begin_inset Formula $\lambda\notin\{\nu_{n}\}_{n}$
+\begin_inset Formula $\psi_{n}(z)\coloneqq(z-a)^{n}$
 \end_inset
 
-, el problema tiene como única solución
-\begin_inset Formula 
-\[
-x(t)=\sum_{n}\frac{1}{\nu_{n}-\lambda}\left(\int_{a}^{b}yu_{n}\right)u_{n}(t),
-\]
+, 
+\begin_inset Formula $(\frac{\psi_{n}}{\Vert\psi_{n}\Vert})_{n}$
+\end_inset
 
+ es una base hilbertiana de 
+\begin_inset Formula $A^{2}(D(a,r))$
 \end_inset
 
-donde la serie converge absoluta y uniformemente para 
-\begin_inset Formula $t\in[a,b]$
+, y el desarrollo en serie de potencias es el desarrollo en serie de Fourier
+ sobre esta base.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
 \end_inset
 
-.
- 
+
 \end_layout
 
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $\lambda=\nu_{k}$
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $\Omega\subsetneq\mathbb{C}$
 \end_inset
 
- para algún 
-\begin_inset Formula $k$
+ es un abierto simplemente conexo y 
+\begin_inset Formula $f:\Omega\to D(0,1)$
 \end_inset
 
-, el problema tiene solución si y sólo si 
-\begin_inset Formula $y\bot u_{k}$
+ es un isomorfismo, 
+\begin_inset Formula $\left(z\mapsto\sqrt{\frac{n}{\pi}}(f(z))^{n-1}\dot{f}(z)\right)_{n}$
 \end_inset
 
-, y entonces las soluciones son
-\begin_inset Formula 
-\begin{align*}
-x(t) & =\alpha u_{k}+\sum_{n\in\mathbb{N}\setminus\{k\}}\frac{1}{\nu_{n}-\lambda}\left(\int_{a}^{b}yu_{n}\right)u_{n}(t), & \alpha & \in\mathbb{C},
-\end{align*}
+ es base hilbertiana de 
+\begin_inset Formula $A^{2}(\Omega)$
+\end_inset
 
+, y en particular para 
+\begin_inset Formula $R>0$
+\end_inset
+
+, 
+\begin_inset Formula $\left(z\mapsto\sqrt{\frac{n}{\pi}}R^{-n}z^{n-1}\right)_{n}$
 \end_inset
 
-donde la serie converge absoluta y uniformemente para 
-\begin_inset Formula $t\in[a,b]$
+ es base hilbertiana de 
+\begin_inset Formula $A^{2}(D(0,R))$
 \end_inset
 
 .
diff --git a/af/n3.lyx b/af/n3.lyx
new file mode 100644
index 0000000..e043d8a
--- /dev/null
+++ b/af/n3.lyx
@@ -0,0 +1,4760 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\usepackage{commath}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Algunos operadores acotados en espacios de Hilbert:
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ espacios prehilbertianos y 
+\begin_inset Formula $G$
+\end_inset
+
+ de dimensión finita con base 
+\begin_inset Formula $(e_{i})_{i}$
+\end_inset
+
+, todo homomorfismo 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ es acotado con
+\begin_inset Formula 
+\[
+\Vert T\Vert\leq\sqrt{\sum_{i}\Vert Te_{i}\Vert^{2}}.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ con bases ortonormales 
+\begin_inset Formula $(e_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(f_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{a_{n}\}_{n}\subseteq\mathbb{K}$
+\end_inset
+
+ una sucesión acotada, el 
+\series bold
+operador diagonal
+\series default
+ 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+T(x)\coloneqq\sum_{n=1}^{\infty}a_{n}\langle x,e_{n}\rangle f_{n}
+\]
+
+\end_inset
+
+es acotado con 
+\begin_inset Formula $\Vert T\Vert=\sup_{n}|a_{n}|$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $g\in L^{\infty}([a,b])$
+\end_inset
+
+, el 
+\series bold
+operador multiplicación por 
+\begin_inset Formula $g$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $T:L^{2}([a,b])\to L^{2}([a,b])$
+\end_inset
+
+ dado por 
+\begin_inset Formula $Tf\coloneqq gf$
+\end_inset
+
+, es acotado con 
+\begin_inset Formula $\Vert T\Vert=\Vert g\Vert_{\infty}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ con bases ortonormales respectivas 
+\begin_inset Formula $(u_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(v_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+ una matriz infinita con 
+\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<\infty$
+\end_inset
+
+, 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+T(x)\coloneqq\sum_{i,j}a_{ij}\langle x,u_{i}\rangle v_{j}
+\]
+
+\end_inset
+
+es un operador acotado con 
+\begin_inset Formula $\Vert T\Vert\leq\sqrt{\sum_{i,j}|a_{ij}|^{2}}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+, el 
+\series bold
+operador integral con núcleo 
+\begin_inset Formula $k$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $K:L^{2}([a,b])\to L^{2}([a,b])$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+K(f)(t)\coloneqq\int_{a}^{b}k(t,s)f(s)\dif s,
+\]
+
+\end_inset
+
+es acotado con 
+\begin_inset Formula $\Vert K\Vert\leq\sqrt{\iint_{[a,b]\times[a,b]}|k|^{2}}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Una matriz infinita 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+ satisface el 
+\series bold
+test de Schur
+\series default
+ si existen 
+\begin_inset Formula $C,D\in\mathbb{R}$
+\end_inset
+
+ tales que
+\begin_inset Formula 
+\begin{align*}
+\forall i\in\mathbb{N},\sum_{j}|a_{ij}| & \leq C, & \forall j\in\mathbb{N}, & \sum_{i}|a_{ij}|\leq D.
+\end{align*}
+
+\end_inset
+
+Entonces, si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ con bases ortonormales respectivas 
+\begin_inset Formula $(u_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(v_{n})_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+T(x)\coloneqq\sum_{i,j}a_{ij}\langle x,u_{i}\rangle v_{j}
+\]
+
+\end_inset
+
+es un operador acotado con 
+\begin_inset Formula $\Vert T\Vert\leq\sqrt{CD}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $k:[a,b]\times[a,b]\to\mathbb{K}$
+\end_inset
+
+ medible y 
+\begin_inset Formula $C,D\in\mathbb{R}$
+\end_inset
+
+ tales que
+\begin_inset Formula 
+\begin{align*}
+\forall t\in[a,b],\int_{a}^{b}|k(t,s)|\dif s & \leq C, & \forall s\in[a,b], & \int_{a}^{b}|k(t,s)|\dif t\leq D,
+\end{align*}
+
+\end_inset
+
+entonces 
+\begin_inset Formula $K:L^{2}([a,b])\to L^{2}([a,b])$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+K(f)(t)\coloneqq\int_{a}^{b}k(t,s)f(s)\dif s
+\]
+
+\end_inset
+
+es un operador acotado con 
+\begin_inset Formula $\Vert K\Vert\leq\sqrt{CD}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ con base ortonormal 
+\begin_inset Formula $(e_{n})_{n}$
+\end_inset
+
+, para 
+\begin_inset Formula $T\in L(H)$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in H$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+T(x)=\sum_{i,j}\langle x,e_{j}\rangle\langle Te_{j},e_{i}\rangle e_{i},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $T$
+\end_inset
+
+ admite una representación matricial 
+\begin_inset Formula $(\langle Te_{j},e_{i}\rangle)_{i,j}\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $T\in L(X,Y)$
+\end_inset
+
+ es 
+\series bold
+de rango finito
+\series default
+ si 
+\begin_inset Formula $\dim\text{Im}T<\infty$
+\end_inset
+
+.
+ Dados espacios de Hilbert 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ y 
+\begin_inset Formula $T\in L(G,H)$
+\end_inset
+
+, 
+\begin_inset Formula $T$
+\end_inset
+
+ es de rango finito si y sólo si viene dada por 
+\begin_inset Formula $T(x)=\sum_{i=1}^{n}\langle x,u_{i}\rangle v_{i}$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $u_{1},\dots,u_{n}\in G$
+\end_inset
+
+ y 
+\begin_inset Formula $v_{1},\dots,v_{n}\in H$
+\end_inset
+
+, en cuyo caso los 
+\begin_inset Formula $(v_{i})_{i}$
+\end_inset
+
+ pueden tomarse de forma que sean una base de 
+\begin_inset Formula $\text{Im}T$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Inversión de operadores
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ son 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios normados, 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+ y 
+\begin_inset Formula $S\in{\cal L}(Y,X)$
+\end_inset
+
+ cumplen 
+\begin_inset Formula $ST=1_{X}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $S$
+\end_inset
+
+ es el 
+\series bold
+inverso por la izquierda
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ y 
+\begin_inset Formula $T$
+\end_inset
+
+ es el 
+\series bold
+inverso por la derecha
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+, y 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+ es 
+\series bold
+invertible
+\series default
+ si existe 
+\begin_inset Formula $T^{-1}\in{\cal L}(Y,X)$
+\end_inset
+
+ inverso de 
+\begin_inset Formula $T$
+\end_inset
+
+ por la izquierda y por la derecha.
+ Llamamos 
+\begin_inset Formula ${\cal L}(X)\coloneqq\text{End}_{\mathbb{K}}X={\cal L}(X,X)$
+\end_inset
+
+ e 
+\begin_inset Formula 
+\[
+\text{Isom}X\coloneqq\text{Isom}_{\mathbb{K}}(X)\coloneqq\{T\in{\cal L}(X)\mid T\text{ invertible}\}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de dimensión finita, 
+\begin_inset Formula $T\in{\cal L}(X)$
+\end_inset
+
+ tiene inverso por la izquierda si y sólo si lo tiene por la derecha, si
+ y sólo si es invertible.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+ Esto no es cierto en general en dimensión infinita; por ejemplo, el operador
+ 
+\series bold
+desplazamiento a derecha
+\series default
+, 
+\begin_inset Formula $S_{\text{r}}\in\ell^{2}$
+\end_inset
+
+ dado por 
+\begin_inset Formula $S_{\text{r}}(x_{1},\dots,x_{n},\dots)\coloneqq(0,x_{1},\dots,x_{n},\dots)$
+\end_inset
+
+, tiene como inverso por la izquierda el 
+\series bold
+desplazamiento a izquierda
+\series default
+, 
+\begin_inset Formula $S_{\text{l}}\in\ell^{2}$
+\end_inset
+
+ dado por 
+\begin_inset Formula $S_{\text{l}}(x_{1},\dots,x_{n},\dots)\coloneqq(x_{2},\dots,x_{n},\dots)$
+\end_inset
+
+, pero no tiene inverso por la derecha.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $T\in\text{End}_{\mathbb{K}}X$
+\end_inset
+
+, 
+\begin_inset Formula $\lambda\in\mathbb{K}$
+\end_inset
+
+ es un 
+\series bold
+valor regular
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ si 
+\begin_inset Formula $T-\lambda1_{X}$
+\end_inset
+
+ es invertible, un 
+\series bold
+valor espectral
+\series default
+ en otro caso, y un 
+\series bold
+valor propio
+\series default
+ si 
+\begin_inset Formula $\ker(T-\lambda1_{X})\neq0$
+\end_inset
+
+, en cuyo caso llamamos 
+\series bold
+subespacio propio
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ correspondiente al valor propio 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ a 
+\begin_inset Formula $\ker(T-\lambda1_{X})$
+\end_inset
+
+ y 
+\series bold
+valores propios
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ correspondientes al valor propio 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ a los elementos no nulos de este subespacio.
+ Llamamos 
+\series bold
+resolvente
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ al conjunto de sus valores regulares, 
+\series bold
+espectro
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma(T)$
+\end_inset
+
+, al conjunto de sus valores espectrales y 
+\series bold
+espectro puntual
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{\text{p}}(T)\subseteq\sigma(T)$
+\end_inset
+
+, al conjunto de sus valores propios.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de dimensión finita, 
+\begin_inset Formula $\sigma_{\text{p}}(T)=\sigma(T)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+ Sin embargo, 
+\begin_inset Formula $0\in\sigma(S_{\text{r}})$
+\end_inset
+
+ pero 
+\begin_inset Formula $\sigma_{\text{p}}(S_{\text{r}})=\emptyset$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach y 
+\begin_inset Formula $T\in{\cal L}(X)$
+\end_inset
+
+ cumple 
+\begin_inset Formula $\Vert T\Vert<1$
+\end_inset
+
+, 
+\begin_inset Formula $1_{X}-T$
+\end_inset
+
+ es invertible con inverso 
+\begin_inset Formula $\sum_{n\in\mathbb{N}}T^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert(1_{X}-T)^{-1}\Vert\leq\frac{1}{1-\Vert T\Vert}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{k=0}^{n}\Vert T^{k}\Vert\leq\sum_{k=0}^{n}\Vert T\Vert^{k}\leq\sum_{k\in\mathbb{N}}\Vert T\Vert^{n}=\frac{1}{1-\Vert T\Vert}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\sum_{n}\Vert T^{n}\Vert$
+\end_inset
+
+ converge y, por ser 
+\begin_inset Formula $X$
+\end_inset
+
+ de Banach, 
+\begin_inset Formula $S\coloneqq\sum_{n}T^{n}$
+\end_inset
+
+ también, pero 
+\begin_inset Formula $S(1_{X}-T)=S-ST=T^{0}=1_{X}$
+\end_inset
+
+ y análogamente 
+\begin_inset Formula $(1_{X}-T)S=1_{X}$
+\end_inset
+
+, luego 
+\begin_inset Formula $S=(1_{X}-T)^{-1}$
+\end_inset
+
+, y finalmente
+\begin_inset Formula 
+\[
+\Vert(1_{X}-T)^{-1}\Vert=\left\Vert \sum_{n}T^{n}\right\Vert \leq\sum_{n}\Vert T\Vert^{n}=\frac{1}{1-\Vert T\Vert}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de von Neumann:
+\series default
+ Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach, 
+\begin_inset Formula $T\in{\cal L}(X)$
+\end_inset
+
+ invertible y 
+\begin_inset Formula $S\in{\cal L}(X)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\Vert T-S\Vert<\frac{1}{\Vert T^{-1}\Vert}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $S$
+\end_inset
+
+ es invertible con
+\begin_inset Formula 
+\begin{align*}
+S^{-1} & =\sum_{n\in\mathbb{N}}(T^{-1}(T-S))^{n}T^{-1}, & \left\Vert T^{-1}-S^{-1}\right\Vert  & \leq\frac{\Vert T^{-1}\Vert^{2}\Vert T-S\Vert}{1-\Vert T^{-1}\Vert\Vert T-S\Vert}.
+\end{align*}
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $\Vert T^{-1}(T-S)\Vert=\Vert T-S\Vert\Vert T^{-1}\Vert<1$
+\end_inset
+
+, luego por el teorema anterior 
+\begin_inset Formula $1_{X}-T^{-1}(T-S)=T^{-1}S$
+\end_inset
+
+ es invertible con
+\begin_inset Formula 
+\[
+(T^{-1}S)^{-1}=\sum_{n}(T^{-1}(T-S))^{n},
+\]
+
+\end_inset
+
+luego 
+\begin_inset Formula $S=T(T^{-1}S)$
+\end_inset
+
+ es invertible con inversa 
+\begin_inset Formula $(T^{-1}S)^{-1}T^{-1}$
+\end_inset
+
+ y
+\begin_inset Formula 
+\begin{align*}
+\Vert T^{-1}-S^{-1}\Vert & =\Vert T^{-1}-(T^{-1}S)^{-1}T^{-1}\Vert=\Vert(1_{X}-(T^{-1}S)^{-1})T^{-1}\Vert\leq\\
+ & \leq\left\Vert \left(1_{X}-\sum_{n}(T^{-1}(T-S))^{n}\right)T^{-1}\right\Vert =\left\Vert \sum_{n\geq1}(T^{-1}(T-S))^{n}T^{-1}\right\Vert \leq\\
+ & \leq\sum_{n\geq1}\Vert(T^{-1}(T-S))^{n}\Vert\Vert T^{-1}\Vert\leq\frac{\Vert T^{-1}\Vert^{2}\Vert T-S\Vert}{1-\Vert T^{-1}\Vert\Vert T-S\Vert}.
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach, 
+\begin_inset Formula $\text{Isom}X$
+\end_inset
+
+ es un abierto de 
+\begin_inset Formula ${\cal L}(X)$
+\end_inset
+
+ y 
+\begin_inset Formula $\cdot^{-1}:\text{Isom}X\to\text{Isom}X$
+\end_inset
+
+ es continua con la norma de 
+\begin_inset Formula ${\cal L}(X)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{FVC}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Liouville:
+\series default
+ Toda función [...][compleja holomorfa y] acotada es constante.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Gelfand:
+\series default
+ Si 
+\begin_inset Formula $_{\mathbb{C}}X$
+\end_inset
+
+ es de Banach y 
+\begin_inset Formula $T\in{\cal L}(X)$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma(T)$
+\end_inset
+
+ es compacto no vacío contenido en 
+\begin_inset Formula $B(0,\Vert T\Vert)$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $\lambda\in\mathbb{C}\setminus B[0,\Vert T\Vert]$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{\Vert T\Vert}{|\lambda|}<1$
+\end_inset
+
+, luego 
+\begin_inset Formula $\lambda1_{X}-T=\lambda(1_{X}-\frac{T}{\lambda})$
+\end_inset
+
+ es invertible y 
+\begin_inset Formula $\lambda\notin\sigma(T)$
+\end_inset
+
+.
+ La función 
+\begin_inset Formula $\psi:\mathbb{C}\to{\cal L}(X)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\psi(\lambda)\coloneqq\lambda1_{X}-T$
+\end_inset
+
+ es continua y por tanto 
+\begin_inset Formula $\mathbb{C}\setminus\sigma(T)=\psi^{-1}(\text{Isom}X)$
+\end_inset
+
+ es abierto, con lo que 
+\begin_inset Formula $\sigma(T)$
+\end_inset
+
+ es cerrado acotado y por tanto compacto.
+ Si fuera vacío, podemos definir 
+\begin_inset Formula $\phi:\mathbb{C}\to\text{Isom}X$
+\end_inset
+
+ como 
+\begin_inset Formula $\phi(\lambda)\coloneqq(\lambda1_{X}-T)^{-1}$
+\end_inset
+
+, que es continua, pero para 
+\begin_inset Formula $\lambda,h\in\mathbb{C}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{multline*}
+\frac{\phi(\lambda+h)-\phi(\lambda)}{h}=\frac{((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1}((\lambda1_{X}-T)-((\lambda+h)1_{X}-T))}{h}=\\
+=-((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1},
+\end{multline*}
+
+\end_inset
+
+de donde
+\begin_inset Formula 
+\[
+\dot{\phi}(\lambda)=\lim_{h\to0}\frac{\phi(\lambda+h)-\phi(\lambda)}{h}=\lim_{h\to0}(-((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1})=-((\lambda1_{X}-T)^{-1})^{2},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es holomorfa y 
+\begin_inset Formula $\dot{\phi}\neq0$
+\end_inset
+
+, pero
+\begin_inset Formula 
+\[
+\Vert\phi(\lambda)\Vert=\Vert(\lambda1_{X}-T)^{-1}\Vert=\frac{1}{|\lambda|}\left\Vert \left(1_{X}-\frac{T}{\lambda}\right)^{-1}\right\Vert =\frac{1}{|\lambda|}\left\Vert \sum_{n\in\mathbb{N}}\frac{T^{n}}{\lambda^{n}}\right\Vert \leq\frac{1}{|\lambda|}\frac{1}{1-\frac{\Vert T\Vert}{|\lambda|}}=\frac{1}{|\lambda|-\Vert T\Vert},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $\lim_{|\lambda|\to\infty}\Vert\phi(\lambda)\Vert=\infty$
+\end_inset
+
+ y por tanto, como 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es continua, es acotada y, por el teorema de Liouville
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Que todavía no hemos visto que se de para espacios vectoriales infinitos
+ pero suponemos que se cumple.
+\end_layout
+
+\end_inset
+
+, 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es constante y 
+\begin_inset Formula $\dot{\phi}=0\#$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+ con 
+\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<1$
+\end_inset
+
+ e 
+\begin_inset Formula $y\in\ell^{2}$
+\end_inset
+
+, el sistema
+\begin_inset Formula 
+\begin{align*}
+x_{k}-\sum_{j\in\mathbb{N}}a_{kj}x_{j} & =y_{k}, & k & \in\mathbb{N},
+\end{align*}
+
+\end_inset
+
+tiene solución única 
+\begin_inset Formula $z\in\ell^{2}$
+\end_inset
+
+, y para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, el sistema truncado
+\begin_inset Formula 
+\begin{align*}
+x_{k}-\sum_{j\in\mathbb{N}_{n}}a_{kj}x_{j} & =y_{k}, & k & \in\mathbb{N}_{n}
+\end{align*}
+
+\end_inset
+
+tiene una única solución 
+\begin_inset Formula $z_{n}\in\mathbb{K}^{n}$
+\end_inset
+
+ de modo que, si 
+\begin_inset Formula $J_{n}:\mathbb{K}^{n}\to\ell^{2}$
+\end_inset
+
+ es la inclusión canónica de 
+\begin_inset Formula $\mathbb{K}^{n}$
+\end_inset
+
+ en las 
+\begin_inset Formula $n$
+\end_inset
+
+ primeras coordenadas, 
+\begin_inset Formula $\lim_{n}J_{n}(z_{n})=z$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert k\Vert_{2}<1$
+\end_inset
+
+ y 
+\begin_inset Formula $g\in L^{2}([a,b])$
+\end_inset
+
+, la ecuación
+\begin_inset Formula 
+\begin{align*}
+f(t)-\int_{a}^{b}k(t,s)f(s)\dif s & =g(t), & t & \in[a,b],
+\end{align*}
+
+\end_inset
+
+tiene solución única que es de la forma
+\begin_inset Formula 
+\[
+g(t)+\int_{a}^{b}\tilde{k}(t,s)g(s)\dif s
+\]
+
+\end_inset
+
+para cierto 
+\begin_inset Formula $\tilde{k}\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $K$
+\end_inset
+
+ es el operador integral con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert k\Vert_{2}<1$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\[
+\forall t\in[a,b],\int_{a}^{b}|k(t,s)|^{2}\dif s\leq C,
+\]
+
+\end_inset
+
+para 
+\begin_inset Formula $g\in L^{2}([a,b])$
+\end_inset
+
+, la serie 
+\begin_inset Formula $\sum_{n}K^{n}g$
+\end_inset
+
+ converge en 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ y converge absoluta y uniformemente en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Con todo esto, para 
+\begin_inset Formula $g\in L^{2}([0,1])$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\in\mathbb{R}\setminus\{1\}$
+\end_inset
+
+, la ecuación integral
+\begin_inset Formula 
+\[
+f(t)-\lambda\int_{0}^{1}\text{e}^{t-s}f(s)\dif s=g(t)
+\]
+
+\end_inset
+
+tiene solución única
+\begin_inset Formula 
+\[
+f(t)=g(t)+\frac{\lambda}{1-\lambda}\int_{0}^{1}\text{e}^{t-s}g(s)\dif s.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Operador adjunto
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son espacios de Hilbert y 
+\begin_inset Formula $T\in L(G,H)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula 
+\[
+\Vert T\Vert=\sup_{x,y\in\overline{B_{G}}}|\langle Tx,y\rangle|=\sup_{x,y\in B_{G}}|\langle Tx,y\rangle|.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Existe un único 
+\begin_inset Formula $T^{*}\in L(H,G)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in G,\forall y\in H,\langle Tx,y\rangle\equiv\langle x,T^{*}y\rangle$
+\end_inset
+
+, el 
+\series bold
+adjunto
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert T\Vert=\Vert T^{*}\Vert$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $H$
+\end_inset
+
+ y 
+\begin_inset Formula $J$
+\end_inset
+
+ 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert, 
+\begin_inset Formula $A,B\in L(G,H)$
+\end_inset
+
+, 
+\begin_inset Formula $C\in L(H,J)$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in\mathbb{K}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(A+B)^{*}=A^{*}+B^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(\alpha A)^{*}=\overline{\alpha}A^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A^{**}=A$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(AC)^{*}=C^{*}A^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es invertible, también lo es 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $(A^{*})^{-1}=(A^{-1})^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert AA^{*}\Vert=\Vert A^{*}A\Vert=\Vert A\Vert^{2}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\ker A=(\text{Im}A^{*})^{\bot}$
+\end_inset
+
+ y 
+\begin_inset Formula $\ker A^{*}=(\text{Im}A)^{\bot}.$
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(\ker A)^{\bot}=\overline{\text{Im}A^{*}}$
+\end_inset
+
+ y 
+\begin_inset Formula $(\ker A^{*})^{\bot}=\overline{\text{Im}A}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+En 
+\begin_inset Formula $\ell^{2}$
+\end_inset
+
+, el adjunto de 
+\begin_inset Formula $S_{\text{r}}$
+\end_inset
+
+ es 
+\begin_inset Formula $S_{\text{l}}$
+\end_inset
+
+ y viceversa.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert y 
+\begin_inset Formula $K\in{\cal L}(H)$
+\end_inset
+
+ es un operador de rango finito dado por 
+\begin_inset Formula $K(x)=\sum_{i=1}^{n}\langle x,u_{i}\rangle v_{i}$
+\end_inset
+
+, su adjunto es de rango finito dado por 
+\begin_inset Formula $K^{*}(x)=\sum_{i=1}^{n}\langle x,v_{i}\rangle u_{i}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert con base 
+\begin_inset Formula $(e_{i})_{i\in I}$
+\end_inset
+
+ y 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ es un operador diagonal con 
+\begin_inset Formula $A(e_{i})\coloneqq\lambda_{i}e_{i}$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $\lambda_{i}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ es un operador diagonal con 
+\begin_inset Formula $A^{*}(e_{i})=\overline{\lambda_{i}}e_{i}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
+\end_inset
+
+ es el operador multiplicación por 
+\begin_inset Formula $g\in L^{\infty}([a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ es el operador multiplicación por 
+\begin_inset Formula $\overline{g}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert separable con base hilbertiana 
+\begin_inset Formula $(e_{n})_{n\in I}$
+\end_inset
+
+ y 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ se expresa en dicha base como 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{I\times I}$
+\end_inset
+
+, 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ se expresa en dicha base como 
+\begin_inset Formula $(\overline{a_{ji}})\in\mathbb{K}^{I\times I}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
+\end_inset
+
+ es el operador integral con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ es el operador integral con núcleo 
+\begin_inset Formula $k^{*}(t,s)\coloneqq\overline{k(s,t)}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert, 
+\begin_inset Formula $M\leq H$
+\end_inset
+
+ es cerrado e 
+\begin_inset Formula $\iota:M\hookrightarrow H$
+\end_inset
+
+ es la inclusión, 
+\begin_inset Formula $\iota^{*}:H\to M$
+\end_inset
+
+ es la proyección ortogonal.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+En general el adjunto no existe en espacios prehilbertianos.
+ Por ejemplo, 
+\begin_inset Formula $T:c_{00}\to c_{00}$
+\end_inset
+
+ dado por 
+\begin_inset Formula $T(x)\coloneqq\sum_{n\geq1}\frac{x_{n}}{n}(1,0,\dots)$
+\end_inset
+
+ no tiene adjunto en 
+\begin_inset Formula $(c_{00},\langle\cdot,\cdot\rangle_{2})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert, 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ es 
+\series bold
+autoadjunto
+\series default
+ o 
+\series bold
+hermitiano
+\series default
+ si 
+\begin_inset Formula $A^{*}=A$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $A,B\in{\cal L}(H)$
+\end_inset
+
+ son autoadjuntos:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert A\Vert=\sup_{x\in\overline{B_{H}}}|\langle Ax,x\rangle|=\sup_{x\in S_{H}}|\langle Ax,x\rangle|$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Los valores propios de 
+\begin_inset Formula $A$
+\end_inset
+
+ son reales.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall x\in H,\langle Ax,x\rangle=0\implies A=0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $H=\ker A\oplus\overline{\text{Im}A}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A+B$
+\end_inset
+
+ es autoadjunto, y 
+\begin_inset Formula $AB$
+\end_inset
+
+ lo es si y sólo si 
+\begin_inset Formula $AB=BA$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $_{\mathbb{C}}H$
+\end_inset
+
+ es un espacio de Hilbert y 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $\forall x\in H,\langle Ax,x\rangle\in\mathbb{R}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\backslash
+Existen únicos 
+\begin_inset Formula $\text{Re}A,\text{Im}A\in{\cal L}(H)$
+\end_inset
+
+ autoadjuntos, la 
+\series bold
+parte real
+\series default
+ y la 
+\series bold
+imaginaria
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+, con 
+\begin_inset Formula $A=\text{Re}A+\text{i}\text{Im}A$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\llbracket A\rrbracket\coloneqq\sup_{x\in S_{H}}|\langle Ax,x\rangle|$
+\end_inset
+
+ es una norma en 
+\begin_inset Formula ${\cal L}(H)$
+\end_inset
+
+ equivalente a la usual.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert con base 
+\begin_inset Formula $(e_{i})_{i\in I}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+El operador diagonal 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ con 
+\begin_inset Formula $T(e_{i})\eqqcolon\lambda_{i}e_{i}$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $\{\lambda_{i}\}_{i\in I}\subseteq\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es separable y 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ se representa respecto a la base como la matriz 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{I\times I}$
+\end_inset
+
+, 
+\begin_inset Formula $A$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $\forall i,j\in I,a_{ij}=\overline{a_{ji}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+El operador multiplicación por 
+\begin_inset Formula $g\in L^{\infty}([a,b])$
+\end_inset
+
+ en 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $g(t)$
+\end_inset
+
+ es real para casi todo 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+El operador integral con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ en 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $k(t,s)=\overline{k(s,t)}$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $(s,t)\in[a,b]\times[a,b]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Una proyección ortogonal 
+\begin_inset Formula $P:H\to H$
+\end_inset
+
+ sobre un subespacio cerrado es autoadjunto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert, 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ es 
+\series bold
+normal
+\series default
+ si 
+\begin_inset Formula $AA^{*}=A^{*}A$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall x,y\in H,\langle Ax,Ay\rangle=\langle A^{*}x,A^{*}y\rangle$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall x\in H,\Vert Ax\Vert=\Vert A^{*}x\Vert$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert complejo, 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ es normal si y sólo si 
+\begin_inset Formula $\text{Re}A\circ\text{Im}A=\text{Im}A\circ\text{Re}A$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Todo operador diagonal es normal.
+\end_layout
+
+\begin_layout Enumerate
+El operador integral sobre 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ es normal si y sólo si
+\begin_inset Formula 
+\[
+\int_{a}^{b}\overline{k(s,t)}k(s,x)\dif s=\int_{a}^{b}k(t,s)\overline{k(x,s)}\dif s
+\]
+
+\end_inset
+
+para casi todo 
+\begin_inset Formula $(t,x)\in[a,b]\times[a,b]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+proyección
+\series default
+ en un espacio normado 
+\begin_inset Formula $X$
+\end_inset
+
+ es un operador 
+\begin_inset Formula $X\to X$
+\end_inset
+
+ idempotente.
+ Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert y 
+\begin_inset Formula $P$
+\end_inset
+
+ es una proyección continua no nula en 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $P$
+\end_inset
+
+ es una proyección ortogonal si y sólo si 
+\begin_inset Formula $\Vert P\Vert=1$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\text{Im}P=(\ker P)^{\bot}$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\ker P=(\text{Im}P)^{\bot}$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $P$
+\end_inset
+
+ es autoadjunto, si y sólo si es normal, si y sólo si 
+\begin_inset Formula $\forall x\in H,\langle Px,x\rangle=\Vert Px\Vert^{2}$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall x\in H,\langle Px,x\rangle\geq0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Existen proyecciones no ortogonales, como 
+\begin_inset Formula $p:\mathbb{R}^{2}\to\mathbb{R}^{2}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $p(x,y)\coloneqq(x+y,0)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\in\mathbb{K}$
+\end_inset
+
+, 
+\begin_inset Formula $\lambda\in\sigma(T)\iff\overline{\lambda}\in\sigma(T^{*})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es normal:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\lambda\in\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $\ker(T-\lambda1_{H})=\ker(T^{*}-\overline{\lambda}1_{H})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\lambda,\mu\in\mathbb{C},(\lambda\neq\mu\implies\ker(T-\lambda1_{H})\bot\ker(T-\mu1_{H}))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\ker(T-\lambda1_{H})$
+\end_inset
+
+ y 
+\begin_inset Formula $\ker(T-\lambda1_{H})^{\bot}$
+\end_inset
+
+ son 
+\begin_inset Formula $T$
+\end_inset
+
+-invariantes.
+\end_layout
+
+\begin_layout Section
+Operadores compactos
+\end_layout
+
+\begin_layout Standard
+Dado un espacio topológico 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $Y\subseteq X$
+\end_inset
+
+ es 
+\series bold
+relativamente compacto
+\series default
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ si su clausura en 
+\begin_inset Formula $X$
+\end_inset
+
+ es compacta.
+ Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios normados, una función lineal 
+\begin_inset Formula $T:X\to Y$
+\end_inset
+
+ es 
+\series bold
+compacta
+\series default
+ si 
+\begin_inset Formula $T(B_{X})$
+\end_inset
+
+ es relativamente compacta en 
+\begin_inset Formula $Y$
+\end_inset
+
+, si y sólo si para cada sucesión acotada 
+\begin_inset Formula $\{x_{n}\}_{n}\subseteq X$
+\end_inset
+
+, 
+\begin_inset Formula $(Tx_{n})_{n}$
+\end_inset
+
+ posee una subsucesión convergente, si y sólo si esto se cumple cuando cada
+ 
+\begin_inset Formula $\Vert x_{n}\Vert=1$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Los operadores de rango finito son compactos.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+El operador identidad en un espacio de dimensión infinita nunca es compacto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula ${\cal K}(X,Y)$
+\end_inset
+
+ al subespacio vectorial de 
+\begin_inset Formula ${\cal L}(X,Y)$
+\end_inset
+
+ de los operadores compactos, que es cerrado si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es de Banach.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $A\in{\cal L}(X,Y)$
+\end_inset
+
+, 
+\begin_inset Formula $T\in{\cal K}(Y,Z)$
+\end_inset
+
+ y 
+\begin_inset Formula $B\in{\cal L}(Z,W)$
+\end_inset
+
+, 
+\begin_inset Formula $BTA\in{\cal K}(X,W)$
+\end_inset
+
+, y en particular 
+\begin_inset Formula ${\cal K}(X)\coloneqq{\cal K}(X,X)$
+\end_inset
+
+ es un ideal de 
+\begin_inset Formula ${\cal L}(X)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $T\in{\cal K}(X,Y)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{Im}T$
+\end_inset
+
+ es un subespacio separable de 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es de Hilbert, 
+\begin_inset Formula $\overline{\text{Im}T}$
+\end_inset
+
+ es de dimensión infinita con base hilbertiana 
+\begin_inset Formula $(e_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ y, para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $P_{n}\in{\cal L}(Y)$
+\end_inset
+
+ es la proyección ortogonal sobre 
+\begin_inset Formula $\text{span}\{e_{i}\}_{i\leq n}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $T=\lim_{n}P_{n}T\in{\cal L}(X,Y)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es de Hilbert, 
+\begin_inset Formula ${\cal K}(X,Y)$
+\end_inset
+
+ es la clausura en 
+\begin_inset Formula ${\cal L}(X,Y)$
+\end_inset
+
+ del conjunto de operadores acotados de rango finito.
+ Esto no es cierto cuando 
+\begin_inset Formula $Y$
+\end_inset
+
+ es un espacio de Banach arbitrario.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son espacios de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(G,H)$
+\end_inset
+
+ es compacto si y sólo si lo es 
+\begin_inset Formula $T^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Standard
+Con esto:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(e_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ y 
+\begin_inset Formula $(f_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ son bases hilbertianas respectivas de 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ y 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ es un operador diagonal dado por 
+\begin_inset Formula $Te_{n}\coloneqq\lambda_{n}f_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $T$
+\end_inset
+
+ es compacto si y sólo si 
+\begin_inset Formula $\lim_{n}\lambda_{n}=0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+El operador multiplicación por 
+\begin_inset Formula $g\in L^{\infty}([a,b])$
+\end_inset
+
+ es compacto si y sólo si 
+\begin_inset Formula $g=0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son espacios de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $T\in{\cal L}(G,H)$
+\end_inset
+
+ se representa en ciertas bases de 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ como 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+, si 
+\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<\infty$
+\end_inset
+
+, 
+\begin_inset Formula $T$
+\end_inset
+
+ es compacto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+El operador integral 
+\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
+\end_inset
+
+ con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ es compacto, 
+\begin_inset Formula ${\cal C}([a,b])$
+\end_inset
+
+ es 
+\begin_inset Formula $K$
+\end_inset
+
+-invariante y 
+\begin_inset Formula $K|_{{\cal C}([a,b])}:({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})\to({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ es compacto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Teorema espectral
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert de dimensión finita y 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es autoadjunto:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\lambda_{1},\dots,\lambda_{m}$
+\end_inset
+
+ son los distintos valores propios de 
+\begin_inset Formula $T$
+\end_inset
+
+, 
+\begin_inset Formula $H=\bigoplus_{k=1}^{m}\ker(T-\lambda_{k}I_{H})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Existe una base ortonormal 
+\begin_inset Formula $(e_{k})_{k}$
+\end_inset
+
+ de 
+\begin_inset Formula $H$
+\end_inset
+
+ formada por vectores propios de 
+\begin_inset Formula $T$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, 
+\begin_inset Formula $Tx=\sum_{k}\mu_{k}\langle x,e_{k}\rangle e_{k}$
+\end_inset
+
+, donde 
+\begin_inset Formula $\mu_{k}$
+\end_inset
+
+ es el valor propio asociado a 
+\begin_inset Formula $e_{k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $T$
+\end_inset
+
+ es un operador compacto autoadjunto en el espacio de Hilbert 
+\begin_inset Formula $H$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert T\Vert$
+\end_inset
+
+ o 
+\begin_inset Formula $-\Vert T\Vert$
+\end_inset
+
+ es valor propio de 
+\begin_inset Formula $T$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Todo operador normal compacto en un 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+-espacio de Hilbert tiene algún valor propio.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es compacto en el 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert 
+\begin_inset Formula $H$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\in\mathbb{K}\setminus0$
+\end_inset
+
+, 
+\begin_inset Formula $\ker(T-\lambda1_{H})$
+\end_inset
+
+ es de dimensión finita.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios de Banach y 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+ compacto, 
+\begin_inset Formula $\sigma_{\text{p}}(T)$
+\end_inset
+
+ es contable, contiene a 
+\begin_inset Formula $\sigma(T)\setminus\{0\}$
+\end_inset
+
+ y, si es infinito, es una sucesión acotada con a lo sumo un punto de acumulació
+n, el 0, y si 
+\begin_inset Formula $T$
+\end_inset
+
+ es normal el 0 es punto de acumulación.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema espectral para operadores compactos autoadjuntos:
+\series default
+ Sean 
+\begin_inset Formula $H$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert y 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ compacto normal:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\sigma_{\text{p}}(T)\setminus\{0\}$
+\end_inset
+
+ es contable.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $P_{\lambda}\in{\cal L}(H)$
+\end_inset
+
+ es la proyección ortogonal sobre 
+\begin_inset Formula $\ker(T-\lambda1_{H})$
+\end_inset
+
+, 
+\begin_inset Formula $T=\sum_{\lambda\in\sigma_{\text{p}}(T)}\lambda P_{\lambda}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\overline{\text{Im}T}=\bigoplus_{\lambda\in\sigma_{\text{p}}(T)\setminus\{0\}}\ker(T-\lambda1_{H})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $H=\ker T\oplus\overline{\text{Im}T}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Existe una base ortonormal 
+\begin_inset Formula $(e_{n})_{n\in J}$
+\end_inset
+
+ de 
+\begin_inset Formula $\overline{\text{Im}T}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{C}$
+\end_inset
+
+ tales que, para 
+\begin_inset Formula $x\in H$
+\end_inset
+
+, 
+\begin_inset Formula $(\mu_{n}\langle x,e_{n}\rangle e_{n})_{n\in J}$
+\end_inset
+
+ es sumable con suma 
+\begin_inset Formula $Tx$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\sigma_{\text{p}}(T)\setminus\{0\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\forall\lambda\in\sigma_{\text{p}}(T)\setminus\{0\},|\{n\in J\mid\mu_{n}=\lambda\}|=\dim\ker(T-\lambda1_{H})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $P_{0}$
+\end_inset
+
+ es la proyección ortogonal sobre 
+\begin_inset Formula $\ker T$
+\end_inset
+
+, 
+\begin_inset Formula $\forall x\in H,x=P_{0}x+\sum_{n\in J}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es compacto autoadjunto si y sólo si hay una familia ortonormal contable
+ 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\end_inset
+
+ de modo que 
+\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+ y 0 es el único punto de acumulación de 
+\begin_inset Formula $(\mu_{n})_{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de alternativa de Fredholm:
+\series default
+ Sean 
+\begin_inset Formula $H$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ compacto autoadjunto, 
+\begin_inset Formula $(e_{n})_{n\in J}$
+\end_inset
+
+ una base ortonormal de 
+\begin_inset Formula $\overline{\text{Im}T}$
+\end_inset
+
+ de modo que 
+\begin_inset Formula $Tx\eqqcolon\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $\mu_{n}\in\mathbb{K}$
+\end_inset
+
+ e 
+\begin_inset Formula $y\in H$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $\lambda\in\mathbb{K}\setminus\{\sigma_{\text{p}}(T)\cup\{0\})$
+\end_inset
+
+, la ecuación 
+\begin_inset Formula $(\lambda1_{H}-T)x=y$
+\end_inset
+
+ tiene como única solución
+\begin_inset Formula 
+\[
+x=\frac{1}{\lambda}\left(y+\sum_{n\in J}\frac{\mu_{n}}{\lambda-\mu_{n}}\langle y,e_{n}\rangle e_{n}\right).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si existe solución 
+\begin_inset Formula $x\in H$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+(\lambda1_{H}-T)x=y\iff\lambda x=Tx+y\iff x=\frac{1}{\lambda}\left(\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}+y\right),
+\]
+
+\end_inset
+
+pero entonces 
+\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\lambda}(\mu_{n}\langle x,e_{n}\rangle+\langle y,e_{n}\rangle)$
+\end_inset
+
+ y 
+\begin_inset Formula $(\lambda-\mu_{n})\langle x,e_{n}\rangle=\langle y,e_{n}\rangle$
+\end_inset
+
+, y como 
+\begin_inset Formula $\lambda-\mu_{n}\neq0$
+\end_inset
+
+, podemos sustituir 
+\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\lambda-\mu_{n}}\langle y,e_{n}\rangle$
+\end_inset
+
+ en lo anterior y queda la solución del enunciado.
+ Queda ver que la serie converge, pero si 
+\begin_inset Formula $\sigma_{\text{p}}(T)$
+\end_inset
+
+ es infinito, 
+\begin_inset Formula $\{\mu_{n}\}_{n}\subseteq\sigma_{\text{p}}(T)$
+\end_inset
+
+ es acotado y por tanto lo es 
+\begin_inset Formula $\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\sum_{n\in J}\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|^{2}|\langle y,e_{n}\rangle|^{2}\leq\sup_{n\in J}\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|^{2}\sum_{n\in J}|\langle y,e_{n}\rangle|^{2}<\infty.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $\lambda\in\sigma_{\text{p}}(T)\setminus\{0\}$
+\end_inset
+
+, la ecuación 
+\begin_inset Formula $(\lambda1_{H}-T)x=y$
+\end_inset
+
+ tiene solución si y sólo si 
+\begin_inset Formula $y\bot\ker(\lambda1_{H}-T)$
+\end_inset
+
+, en cuyo caso las soluciones son
+\begin_inset Formula 
+\begin{align*}
+x & =\frac{1}{\lambda}\left(y+\sum_{\begin{subarray}{c}
+n\in J\\
+\mu_{n}\neq\lambda
+\end{subarray}}\frac{\mu_{n}}{\lambda-\mu_{n}}\langle y,e_{n}\rangle e_{n}\right)+z, & z & \in\ker(\lambda1_{H}-T).
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si la ecuación tiene solución 
+\begin_inset Formula $x$
+\end_inset
+
+, entonces 
+\begin_inset Formula $y=(\lambda1_{H}-T)x\in\text{Im}(\lambda1_{H}-T)\subseteq\overline{\text{Im}(\lambda1_{H}-T)}=\ker((\lambda1_{H}-T)^{*})^{\bot}=\ker(\lambda1_{H}-T)^{\bot}$
+\end_inset
+
+ por ser 
+\begin_inset Formula $1_{H}$
+\end_inset
+
+ y 
+\begin_inset Formula $T$
+\end_inset
+
+ autoadjuntos, y claramente dos soluciones difieren en un vector de 
+\begin_inset Formula $\ker(\lambda1_{H}-T)$
+\end_inset
+
+.
+ Queda ver que, si 
+\begin_inset Formula $y\in\ker(\lambda1_{H}-T)^{\bot}$
+\end_inset
+
+, la 
+\begin_inset Formula $x$
+\end_inset
+
+ del enunciado es solución, para lo cual hacemos la misma sustitución que
+ al principio del primer apartado pero, cuando 
+\begin_inset Formula $\lambda=\mu_{n}$
+\end_inset
+
+, en su lugar vemos que 
+\begin_inset Formula $(\lambda-\mu_{n})\langle x,e_{n}\rangle=\langle y,e_{n}\rangle$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\langle y,e_{n}\rangle=0$
+\end_inset
+
+, por lo que excluimos dicho factor de la serie, la cual converge por el
+ mismo motivo que en el primer apartado y resulta en la solución del enunciado.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $y=0$
+\end_inset
+
+, 
+\begin_inset Formula $Tx=y$
+\end_inset
+
+ tiene solución si y sólo si 
+\begin_inset Formula $y\bot\ker T$
+\end_inset
+
+ y 
+\begin_inset Formula $\sum_{n\in J}\left|\frac{\langle y,e_{n}\rangle}{\mu_{n}}\right|^{2}<\infty$
+\end_inset
+
+, en cuyo caso las soluciones son
+\begin_inset Formula 
+\begin{align*}
+x & =\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+z, & z & \in\ker T.
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si la ecuación tiene solución 
+\begin_inset Formula $x$
+\end_inset
+
+, 
+\begin_inset Formula $y\in\text{Im}T\subseteq(\ker T)^{\bot}$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}=Tx=y=\sum_{n\in J}\langle y,e_{n}\rangle e_{n},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\mu_{n}}\langle y,e_{n}\rangle$
+\end_inset
+
+ para cada 
+\begin_inset Formula $n$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\sum_{n\in J}\left|\frac{\langle y,e_{n}\rangle}{\mu_{n}}\right|^{2}=\Vert x\Vert^{2}<\infty$
+\end_inset
+
+, y como 
+\begin_inset Formula $(e_{n})_{n}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $\overline{\text{Im}T}$
+\end_inset
+
+, 
+\begin_inset Formula $x\in\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+\overline{\text{Im}T}^{\bot}$
+\end_inset
+
+ con 
+\begin_inset Formula $\overline{\text{Im}T}^{\bot}=\ker T$
+\end_inset
+
+.
+ Finalmente, si esta condición se cumple, 
+\begin_inset Formula $y\in\overline{\text{Im}T}$
+\end_inset
+
+, la serie del enunciado converge y
+\begin_inset Formula 
+\[
+T\left(\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+z\right)=\sum_{n\in J}\langle y,e_{n}\rangle e_{n}+0=y.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Sea 
+\begin_inset Formula $A$
+\end_inset
+
+ un operador en un espacio de Hilbert 
+\begin_inset Formula $H$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A$
+\end_inset
+
+ es una isometría si y sólo si 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ es inverso por la izquierda de 
+\begin_inset Formula $A$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall x,y\in H,\langle Ax,Ay\rangle=\langle x,y\rangle$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A$
+\end_inset
+
+ es un isomorfismo isométrico, si y sólo si es una isometría suprayectiva,
+ si y sólo si 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ es inverso de 
+\begin_inset Formula $A$
+\end_inset
+
+, y entonces decimos que 
+\begin_inset Formula $A$
+\end_inset
+
+ es 
+\series bold
+unitario
+\series default
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $H$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert y 
+\begin_inset Formula $S,T\in{\cal L}(H)$
+\end_inset
+
+ compactos autoadjuntos, 
+\begin_inset Formula $\forall\lambda\in\mathbb{K},\dim\ker(T-\lambda1_{H})=\dim\ker(S-\lambda1_{H})$
+\end_inset
+
+ si y sólo si existe 
+\begin_inset Formula $U\in{\cal L}(H)$
+\end_inset
+
+ unitario con 
+\begin_inset Formula $U^{*}SU=T$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $S,T\in{\cal L}(H)$
+\end_inset
+
+ en el 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert 
+\begin_inset Formula $H$
+\end_inset
+
+ son 
+\series bold
+simultáneamente diagonalizables
+\series default
+ si existe una familia ortonormal 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\alpha_{n}\}_{n\in J},\{\beta_{n}\}_{n\in J}\subseteq\mathbb{K}$
+\end_inset
+
+ tal que
+\begin_inset Formula 
+\[
+\forall x\in H,\left(Sx=\sum_{n\in J}\alpha_{n}\langle x,e_{n}\rangle e_{n}\land Tx=\sum_{n\in J}\beta_{n}\langle x,e_{n}\rangle e_{n}\right).
+\]
+
+\end_inset
+
+Si 
+\begin_inset Formula $S$
+\end_inset
+
+ y 
+\begin_inset Formula $T$
+\end_inset
+
+ son compactos y autoadjuntos esto equivale a que 
+\begin_inset Formula $ST=TS$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema espectral para operadores compactos normales:
+\series default
+ Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+-espacio de Hilbert y 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ compacto normal, ocurre lo mismo que en el anterior teorema espectral.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+-espacio de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es compacto normal si y sólo si hay una familia ortonormal contable 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{C}$
+\end_inset
+
+ con 0 como único punto de acumulación de modo que 
+\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un operador entre 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert 
+\begin_inset Formula $T\in{\cal L}(G,H)$
+\end_inset
+
+ es compacto si y sólo si hay una familia contable 
+\begin_inset Formula $\{\nu_{n}\}_{n\in J}\subseteq\mathbb{R}^{+}$
+\end_inset
+
+ con 0 como punto de acumulación, 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq G$
+\end_inset
+
+ y 
+\begin_inset Formula $\{f_{n}\}_{n\in J}\subseteq H$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\nu_{n}\langle x,e_{n}\rangle f_{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Ecuaciones integrales de Fredholm
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+ecuación integral de Fredholm
+\series default
+ es una de la forma
+\begin_inset Formula 
+\[
+x(t)-\mu\int_{a}^{b}k(t,s)x(s)\dif s=g(t),
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $x,g\in L^{2}([a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ y la incógnita es 
+\begin_inset Formula $x$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Un núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ es 
+\series bold
+simétrico
+\series default
+ si 
+\begin_inset Formula $k(t,s)=\overline{k(s,t)}$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $s,t\in[a,b]$
+\end_inset
+
+.
+ 
+\series bold
+Teorema de alternativa de Fredholm:
+\series default
+ Sean 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ un núcleo simétrico, 
+\begin_inset Formula $K$
+\end_inset
+
+ el operador integral asociado y 
+\begin_inset Formula $g\in L^{2}([a,b])$
+\end_inset
+
+, si 
+\begin_inset Formula $Kx=\sum_{n\in J}\mu_{j}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+ para cierta base hilbertiana contable 
+\begin_inset Formula $(e_{n})_{n\in J}$
+\end_inset
+
+ de 
+\begin_inset Formula $\overline{\text{Im}K}$
+\end_inset
+
+, ciertos 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\end_inset
+
+ y todo 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, considerando la ecuación integral de Fredholm de arriba, 
+\begin_inset Formula $x-Kx=g$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\mu=0$
+\end_inset
+
+, la ecuación tiene como única solución 
+\begin_inset Formula $x=g$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\frac{1}{\mu}\notin\{\mu_{n}\}_{n}$
+\end_inset
+
+, la ecuación tiene como única solución
+\begin_inset Formula 
+\[
+x(t)=g(t)+\mu\left(\sum_{n}\frac{\mu_{n}}{1-\mu\mu_{n}}\left(\int_{a}^{b}g\overline{e_{n}}\right)e_{n}(t)\right),
+\]
+
+\end_inset
+
+y existe 
+\begin_inset Formula $\alpha>0$
+\end_inset
+
+ que depende solo de 
+\begin_inset Formula $k$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\Vert x\Vert_{2}\leq\alpha\Vert g\Vert_{2}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si existe 
+\begin_inset Formula $n\in J$
+\end_inset
+
+ con 
+\begin_inset Formula $\mu_{n}=\frac{1}{\mu}$
+\end_inset
+
+, la ecuación tiene solución si y sólo si 
+\begin_inset Formula $g\bot\ker(\frac{1_{L^{2}([a,b])}}{\mu}-K)$
+\end_inset
+
+, y entonces las soluciones son
+\begin_inset Formula 
+\begin{align*}
+x(t) & =g(t)+\mu\sum_{\begin{subarray}{c}
+n\in J\\
+\mu_{n}\neq\frac{1}{\mu}
+\end{subarray}}\frac{\mu_{n}}{1-\mu\mu_{n}}\left(\int g\overline{e_{n}}\right)e_{j}+u, & u & \in\ker(\tfrac{1_{L^{2}([a,b])}}{\mu}-K).
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La convergencia de las series es de media cuadrática, pero en ciertos casos
+ puede ser uniforme.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ es un núcleo simétrico con
+\begin_inset Formula 
+\[
+\sup_{t\in[a,b]}\int_{a}^{b}|k(t,s)|^{2}\dif s<\infty,
+\]
+
+\end_inset
+
+
+\begin_inset Formula $K$
+\end_inset
+
+ es el operador integral asociado y hay una base hilbertiana 
+\begin_inset Formula $(e_{n})_{n\in J}$
+\end_inset
+
+ de 
+\begin_inset Formula $\overline{\text{Im}K}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\end_inset
+
+ y tales que 
+\begin_inset Formula $Kx=\sum_{n}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Teorema de Hilbert-Schmidt:
+\series default
+ Para 
+\begin_inset Formula $x\in L^{2}([a,b])$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\int_{a}^{b}k(t,s)x(s)\dif s=\sum_{n\in J}\mu_{n}\left(\int_{a}^{b}x\overline{e_{n}}\right)e_{n}(t)
+\]
+
+\end_inset
+
+para casi todo 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+, y si 
+\begin_inset Formula $J$
+\end_inset
+
+ es numerable la serie converge absoluta y uniformemente en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para la primera parte basta tomar en el teorema anterior un 
+\begin_inset Formula $\mu\neq0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\frac{1}{\mu}$
+\end_inset
+
+ no sea valor propio y despejar.
+ Para la segunda podemos suponer 
+\begin_inset Formula $J=(\mathbb{N},\geq)$
+\end_inset
+
+, y queremos ver que
+\begin_inset Formula 
+\[
+\sum_{n}\left|\mu_{n}\left(\int_{a}^{b}x\overline{e_{n}}\right)e_{n}(t)\right|=\sum_{n}|\mu_{n}\langle x,e_{n}\rangle e_{n}(t)|
+\]
+
+\end_inset
+
+es uniformemente de Cauchy en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+ Por la desigualdad de Cauchy-Schwartz,
+\begin_inset Formula 
+\[
+\sum_{n=p}^{q}|\mu_{n}e_{n}(t)||\langle x,e_{n}\rangle|\leq\sqrt{\sum_{n=p}^{q}|\mu_{n}e_{n}(t)|^{2}\sum_{n=p}^{q}|\langle x,e_{n}\rangle|^{2}},
+\]
+
+\end_inset
+
+pero para 
+\begin_inset Formula $n\in J$
+\end_inset
+
+ y 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\mu_{n}e_{n}(t)=K(e_{n})(t)=\int_{a}^{b}k(t,s)e_{k}(s)\dif s=\langle e_{k},\overline{k_{t}}\rangle,
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $k_{t}(s)\coloneqq k(t,s)$
+\end_inset
+
+, luego
+\begin_inset Formula 
+\[
+\sqrt{\sum_{n=p}^{q}|\mu_{n}e_{n}(t)|^{2}}=\sqrt{\sum_{n=p}^{q}|\langle e_{n},\overline{k_{t}}\rangle|^{2}}\leq\Vert k_{t}\Vert_{2}\leq\sup_{t\in[a,b]}\Vert k_{t}\Vert_{2}<\infty,
+\]
+
+\end_inset
+
+con lo que esto está acotado superiormente por un valor independiente de
+ 
+\begin_inset Formula $t$
+\end_inset
+
+ y el resultado sale de que 
+\begin_inset Formula $|\langle x,e_{n}\rangle|^{2}$
+\end_inset
+
+ tampoco depende de 
+\begin_inset Formula $t$
+\end_inset
+
+ y 
+\begin_inset Formula $\lim_{p,q}\sum_{n=p}^{q}|\langle x,e_{n}\rangle|^{2}=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Las series del teorema de alternativa de Fredholm convergen absoluta y uniformem
+ente en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $k\in{\cal C}([a,b]\times[a,b])$
+\end_inset
+
+ es un núcleo simétrico, existen una familia ortonormal contable 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq({\cal C}([a,b]),\Vert\cdot\Vert_{2})$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\end_inset
+
+ tales que, si 
+\begin_inset Formula $K$
+\end_inset
+
+ es el operador integral asociado a 
+\begin_inset Formula $k$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in{\cal C}([a,b])$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+Kf(t)=\sum_{n\in J}\mu_{n}\left(\int_{a}^{b}f\overline{e_{n}}\right)e_{n}(t)
+\]
+
+\end_inset
+
+para todo 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+ y la convergencia de la serie es absoluta y uniforme.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Problemas de Sturm-Liouville
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+problema regular de Sturm-Liouville
+\series default
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+La forma general del problema tiene como ecuación 
+\begin_inset Formula $\od{}{x}(p\dot{x})+qx+\lambda\sigma x+y=0$
+\end_inset
+
+ con 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ continuas y estrictamente positivas.
+ Aquí tomamos 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $q$
+\end_inset
+
+ constantes en 1.
+\end_layout
+
+\end_inset
+
+ es uno de la forma
+\begin_inset Formula 
+\begin{align*}
+-\ddot{x}+qx-\lambda x & =y, & \alpha x(a)+\beta\dot{x}(a) & =0, & \gamma x(b)+\delta\dot{x}(b) & =0,
+\end{align*}
+
+\end_inset
+
+donde 
+\begin_inset Formula $q\in{\cal C}([a,b],\mathbb{R})$
+\end_inset
+
+, 
+\begin_inset Formula $y\in{\cal C}([a,b],\mathbb{C})$
+\end_inset
+
+, 
+\begin_inset Formula $\lambda\in\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha,\beta,\gamma,\delta\in\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $|\alpha|+|\beta|,|\gamma|+|\delta|\neq0$
+\end_inset
+
+ y la incógnita 
+\begin_inset Formula $x\in{\cal C}^{2}([a,b],\mathbb{C})$
+\end_inset
+
+.
+ Su 
+\series bold
+operador de Sturm-Liouville
+\series default
+ asociado es 
+\begin_inset Formula $S\in{\cal L}(D_{S},{\cal C}([a,b],\mathbb{C}))$
+\end_inset
+
+ dado por 
+\begin_inset Formula $S(x)\coloneqq-\ddot{x}+qx$
+\end_inset
+
+, donde
+\begin_inset Formula 
+\[
+D_{S}\coloneqq\{x\in{\cal C}^{2}([a,b],\mathbb{C})\mid\alpha x(a)+\beta\dot{x}(a)=\gamma x(b)+\delta\dot{x}(b)=0\},
+\]
+
+\end_inset
+
+y entonces el problema anterior es 
+\begin_inset Formula $(S-\mu1_{D_{S}})x=y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $q\in{\cal C}([a,b],\mathbb{R})$
+\end_inset
+
+ e 
+\begin_inset Formula $y_{0},y_{1}\in\mathbb{R}$
+\end_inset
+
+, el problema de Cauchy
+\begin_inset Formula 
+\begin{align*}
+-\ddot{x}+qx & =0, & x(a) & =y_{0}, & \dot{x}(a) & =y_{1}
+\end{align*}
+
+\end_inset
+
+tiene una única solución real, y para 
+\begin_inset Formula $\alpha,\beta\in\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $|\alpha|+|\beta|\neq0$
+\end_inset
+
+, si 
+\begin_inset Formula $(y_{0},y_{1})\in\mathbb{R}^{2}$
+\end_inset
+
+ recorre la recta 
+\begin_inset Formula $\alpha y_{0}+\beta y_{1}=0$
+\end_inset
+
+, la correspondiente solución del problema recorre una recta (subespacio
+ de dimensión 1) de 
+\begin_inset Formula ${\cal C}^{2}([a,b])$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+determinante wronskiano
+\series default
+ de 
+\begin_inset Formula $x_{1},\dots,x_{n}\in{\cal C}^{n-1}([a,b],\mathbb{K})$
+\end_inset
+
+ es 
+\begin_inset Formula $W(x_{1},\dots,x_{n}):[a,b]\to\mathbb{K}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $t\mapsto\det(x_{j}^{(i)}(t))_{0\leq i<n}^{1\leq j\leq n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $S:D_{S}\to{\cal C}([a,b],\mathbb{C})$
+\end_inset
+
+ es un operador de Sturm-Liouville asociado al problema con parámetros 
+\begin_inset Formula $q,y,\lambda,\alpha,\beta,\gamma,\delta$
+\end_inset
+
+, existen 
+\begin_inset Formula $u,v\in{\cal C}([a,b],\mathbb{R})$
+\end_inset
+
+ con 
+\begin_inset Formula $-\ddot{u}+qu=0$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha x(a)+\beta\dot{x}(a)=0$
+\end_inset
+
+, 
+\begin_inset Formula $-\ddot{v}+qv=0$
+\end_inset
+
+ y 
+\begin_inset Formula $\gamma x(b)+\delta\dot{x}(b)=0$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $W(u,v)(t)$
+\end_inset
+
+ es constante en 
+\begin_inset Formula $t$
+\end_inset
+
+ y, si 
+\begin_inset Formula $S$
+\end_inset
+
+ es inyectivo, 
+\begin_inset Formula $W(u,v)(t)\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $u$
+\end_inset
+
+ y 
+\begin_inset Formula $v$
+\end_inset
+
+ son linealmente independientes, y llamamos 
+\series bold
+función de Green
+\series default
+ asociada a 
+\begin_inset Formula $S$
+\end_inset
+
+ al núcleo simétrico 
+\begin_inset Formula $k\in{\cal C}([a,b]\times[a,b])$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+k(t,s)\coloneqq-\frac{u(\min\{t,s\})v(\max\{t,s\})}{W(u,v)(a)},
+\]
+
+\end_inset
+
+que no depende de 
+\begin_inset Formula $u$
+\end_inset
+
+ y 
+\begin_inset Formula $v$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $S:D_{S}\to{\cal C}([a,b])$
+\end_inset
+
+ es un operador de Sturm-Liouville inyectivo con función de Green 
+\begin_inset Formula $k$
+\end_inset
+
+, llamamos 
+\series bold
+operador de Green
+\series default
+ asociado a 
+\begin_inset Formula $S$
+\end_inset
+
+ al operador integral 
+\begin_inset Formula $G:L^{2}([a,b])\to L^{2}([a,b])$
+\end_inset
+
+ asociado al núcleo 
+\begin_inset Formula $k$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $G|_{{\cal C}([a,b])}$
+\end_inset
+
+ es el inverso de 
+\begin_inset Formula $S$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, 
+\begin_inset Formula $(S-\mu1_{D_{S}})x=y$
+\end_inset
+
+ tiene solución única 
+\begin_inset Formula $x\in D_{S}$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $(1_{{\cal C}([a,b])}-\mu G)x=Gy$
+\end_inset
+
+ tiene solución única 
+\begin_inset Formula $x\in{\cal C}([a,b])$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $S:D_{S}\to{\cal C}([a,b],\mathbb{C})$
+\end_inset
+
+ es el operador de Sturm-Liouville asociado al problema con parámetros 
+\begin_inset Formula $q,y,\lambda,\alpha,\beta,\gamma,\delta$
+\end_inset
+
+, existe una sucesión 
+\begin_inset Formula $(\nu_{n})_{n}$
+\end_inset
+
+ de reales distintos con 
+\begin_inset Formula $\sum_{n}\frac{1}{\nu_{n}^{2}}<\infty$
+\end_inset
+
+ y una base hilbertiana numerable 
+\begin_inset Formula $(u_{n})_{n}$
+\end_inset
+
+ de 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ tales que:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall n\in\mathbb{N},Su_{n}=\nu_{n}u_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula 
+\[
+\forall x\in D_{S},\forall t\in[a,b],x(t)=\sum_{n}\left(\int_{a}^{b}xu_{n}\right)u_{n}(t),
+\]
+
+\end_inset
+
+donde la serie converge absoluta y uniformemente para 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\lambda\notin\{\nu_{n}\}_{n}$
+\end_inset
+
+, el problema tiene como única solución
+\begin_inset Formula 
+\[
+x(t)=\sum_{n}\frac{1}{\nu_{n}-\lambda}\left(\int_{a}^{b}yu_{n}\right)u_{n}(t),
+\]
+
+\end_inset
+
+donde la serie converge absoluta y uniformemente para 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\lambda=\nu_{k}$
+\end_inset
+
+ para algún 
+\begin_inset Formula $k$
+\end_inset
+
+, el problema tiene solución si y sólo si 
+\begin_inset Formula $y\bot u_{k}$
+\end_inset
+
+, y entonces las soluciones son
+\begin_inset Formula 
+\begin{align*}
+x(t) & =\alpha u_{k}+\sum_{n\in\mathbb{N}\setminus\{k\}}\frac{1}{\nu_{n}-\lambda}\left(\int_{a}^{b}yu_{n}\right)u_{n}(t), & \alpha & \in\mathbb{C},
+\end{align*}
+
+\end_inset
+
+donde la serie converge absoluta y uniformemente para 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/af/n4.lyx b/af/n4.lyx
new file mode 100644
index 0000000..95113e8
--- /dev/null
+++ b/af/n4.lyx
@@ -0,0 +1,6992 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\usepackage{commath}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
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+\graphics default
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+\spacing single
+\use_hyperref false
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+\use_package amsmath 1
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+\use_package mathdots 1
+\use_package mathtools 1
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+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
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+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Los 
+\series bold
+principios fundamentales del análisis funcional
+\series default
+ son el teorema de Hahn-Banach, el teorema de la acotación uniforme y el
+ teorema de la gráfica cerrada.
+\end_layout
+
+\begin_layout Section
+Teorema de Hahn-Banach
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Tychonoff:
+\series default
+ Si 
+\begin_inset Formula $(X_{i})_{i\in I}$
+\end_inset
+
+ son espacios topológicos compactos, 
+\begin_inset Formula $\prod_{i\in I}X_{i}$
+\end_inset
+
+ es compacto con la topología producto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de extensión de Hann-Banach:
+\series default
+ Sean 
+\begin_inset Formula $Y\leq_{\mathbb{K}}X$
+\end_inset
+
+, 
+\begin_inset Formula $p:X\to\mathbb{R}$
+\end_inset
+
+ subaditiva y positivamente homogénea y 
+\begin_inset Formula $f:Y\to\mathbb{K}$
+\end_inset
+
+ lineal con 
+\begin_inset Formula $f\leq p|_{Y}$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ se extiende a 
+\begin_inset Formula $\hat{f}:X\to\mathbb{R}$
+\end_inset
+
+ lineal con 
+\begin_inset Formula $\hat{f}\leq p$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+ 
+\series bold
+Demostración
+\series default
+ para 
+\begin_inset Formula $Y$
+\end_inset
+
+ de codimensión 1
+\series bold
+:
+\series default
+ Sea 
+\begin_inset Formula $x_{0}\in X\setminus Y$
+\end_inset
+
+, entonces 
+\begin_inset Formula $X=Y\oplus\text{span}\{x_{0}\}$
+\end_inset
+
+ y toda extensión lineal 
+\begin_inset Formula $\hat{f}:X\to\mathbb{R}$
+\end_inset
+
+ se escribe como 
+\begin_inset Formula $\hat{f}(y+ax_{0})=f(y)+a\hat{f}(x_{0})$
+\end_inset
+
+ para cada 
+\begin_inset Formula $y+ax_{0}\in X$
+\end_inset
+
+ con 
+\begin_inset Formula $y\in Y$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+, y queremos ver que existe 
+\begin_inset Formula $\alpha\in\mathbb{R}$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $\hat{f}(x_{0})=\alpha$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\hat{f}\leq p$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $a=0$
+\end_inset
+
+ esto siempre se cumple; para 
+\begin_inset Formula $a>0$
+\end_inset
+
+
+\begin_inset Formula 
+\begin{multline*}
+\forall y\in Y,\hat{f}(y+ax_{0})=f(y)+a\alpha\leq p(y+ax_{0})\iff\forall y\in Y,f\left(\frac{y}{a}\right)+\alpha\leq p\left(\frac{y}{a}+x_{0}\right)\iff\\
+\iff\forall z\in Y,\alpha\leq-f(z)+p(z+x_{0}),
+\end{multline*}
+
+\end_inset
+
+y para 
+\begin_inset Formula $a<0$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{multline*}
+\forall y\in Y,\hat{f}(y+ax_{0})=f(y)+a\alpha\leq p(y+ax_{0})\iff\forall y\in Y,f\left(-\frac{y}{a}\right)-\alpha\leq p\left(-\frac{y}{a}-x_{0}\right)\iff\\
+\iff\forall w\in Y,\alpha\geq f(w)-p(w-x_{0}),
+\end{multline*}
+
+\end_inset
+
+con lo que la condición equivale a que 
+\begin_inset Formula $\forall z,w\in Y,f(w)-p(w-x_{0})\leq\alpha\leq-f(z)+p(z+x_{0})$
+\end_inset
+
+, pero siempre existe tal 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ ya que, para 
+\begin_inset Formula $z,w\in Y$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+f(z)+f(w)=f(z+w)\leq p(z+w)=p(z+x_{0}+w-x_{0})\leq p(z+x_{0})+p(w-x_{0}).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+El teorema de Tychonoff equivale al axioma de elección y es estrictamente
+ más fuerte que el teorema de Tychonoff para espacios compactos separados,
+ el cual implica el teorema de extensión de Hann-Banach.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Hann-Banach (
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+) y Sobczyk (
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+):
+\series default
+ Sean 
+\begin_inset Formula $Y\leq_{\mathbb{K}}X$
+\end_inset
+
+, 
+\begin_inset Formula $p:X\to\mathbb{K}$
+\end_inset
+
+ una seminorma y 
+\begin_inset Formula $f:Y\to\mathbb{K}$
+\end_inset
+
+ lineal con 
+\begin_inset Formula $|f|\leq p|_{Y}$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ se extiende a una 
+\begin_inset Formula $\hat{f}:X\to\mathbb{K}$
+\end_inset
+
+ lineal con 
+\begin_inset Formula $|\hat{f}|\leq p$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio normado e 
+\begin_inset Formula $Y\leq X$
+\end_inset
+
+, toda 
+\begin_inset Formula $f\in Y^{*}$
+\end_inset
+
+ se extiende a una 
+\begin_inset Formula $\hat{f}\in X^{*}$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert\hat{f}\Vert=\Vert f\Vert$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $p:X\to\mathbb{R}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $p(x)\coloneqq\Vert f\Vert\Vert x\Vert$
+\end_inset
+
+ es subaditiva y positivamente homogénea con 
+\begin_inset Formula $|f(x)|\leq\Vert f\Vert\Vert x\Vert=p(x)$
+\end_inset
+
+, luego 
+\begin_inset Formula $f$
+\end_inset
+
+ se extiende a 
+\begin_inset Formula $\hat{f}:X\to\mathbb{R}$
+\end_inset
+
+ lineal con 
+\begin_inset Formula $|\hat{f}|\leq p$
+\end_inset
+
+ y, para 
+\begin_inset Formula $x\in S_{X}$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert\hat{f}(x)\Vert\leq\Vert f\Vert$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $\Vert f\Vert\leq\Vert\hat{f}\Vert\leq\Vert f\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema de Hann-Banach
+\series default
+ es el anterior cuando 
+\begin_inset Formula $X$
+\end_inset
+
+ es real y separable.
+ 
+\series bold
+Demostración
+\series default
+ sin usar cosas de esta sección no probadas
+\series bold
+:
+\series default
+ Sean 
+\begin_inset Formula $\{x_{n}\}_{n\in\mathbb{N}}$
+\end_inset
+
+ denso en 
+\begin_inset Formula $X$
+\end_inset
+
+ y, para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $X_{n}\coloneqq\text{span}\{Y\cup\{x_{k}\}_{k\in\mathbb{N}_{n}}\}$
+\end_inset
+
+, o 
+\begin_inset Formula $X_{n}=X_{n+1}$
+\end_inset
+
+ o es un subespacio de 
+\begin_inset Formula $X_{n+1}$
+\end_inset
+
+ de codimensión 1, y por inducción en lo anterior para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ existe 
+\begin_inset Formula $f_{n}\in X_{n}^{*}$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert f_{n}\Vert=\Vert f\Vert$
+\end_inset
+
+ y 
+\begin_inset Formula $f_{n}=f_{n+1}|_{X_{n}^{*}}$
+\end_inset
+
+, de modo que si 
+\begin_inset Formula $Z\coloneqq\bigcup_{n}X_{n}$
+\end_inset
+
+, existe 
+\begin_inset Formula $F\in Z^{*}$
+\end_inset
+
+ con 
+\begin_inset Formula $f=F|_{Y}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert F\Vert=\Vert f\Vert$
+\end_inset
+
+, pero para 
+\begin_inset Formula $y\in X$
+\end_inset
+
+ existe 
+\begin_inset Formula $\{z_{n}\}_{n}\subseteq Z$
+\end_inset
+
+ convergente a 
+\begin_inset Formula $y$
+\end_inset
+
+ y, por continuidad de 
+\begin_inset Formula $F$
+\end_inset
+
+, existe 
+\begin_inset Formula $\hat{f}(y)\coloneqq\lim_{n}F(y_{n})$
+\end_inset
+
+, con 
+\begin_inset Formula $\hat{f}(y)$
+\end_inset
+
+ independiente de la sucesión elegida, con lo que podemos definir 
+\begin_inset Formula $\hat{f}:X\to\mathbb{R}$
+\end_inset
+
+ de esta forma y claramente es lineal y continua con 
+\begin_inset Formula $\Vert\hat{f}\Vert=\Vert F\Vert=\Vert f\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea entonces 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio normado:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall x\in X\setminus0,\exists f\in X^{*}:(\Vert f\Vert=1\land f(x)=\Vert x\Vert)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall x\in X,\Vert x\Vert=\max_{f\in B_{X^{*}}}|f(x)|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $Y\leq X$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ con 
+\begin_inset Formula $\delta\coloneqq d(x,Y)>0$
+\end_inset
+
+, 
+\begin_inset Formula $\exists f\in X^{*}:(f(Y)=0\land f(x)=1\land\Vert f\Vert=\delta^{-1})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula 
+\[
+\forall Y\leq X,\overline{Y}=\bigcap_{\begin{subarray}{c}
+f\in X^{*}\\
+Y\subseteq\ker f
+\end{subarray}}\ker f.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula 
+\[
+\forall S\subseteq X,\overline{\text{span}S}\coloneqq\bigcap_{\begin{subarray}{c}
+f\in X^{*}\\
+S\subseteq\ker f
+\end{subarray}}\ker f.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $S\subseteq X$
+\end_inset
+
+ es total si y sólo si 
+\begin_inset Formula $\forall f\in X^{*},(f(S)=0\implies f=0)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $x_{1},\dots,x_{n}\in X$
+\end_inset
+
+ son linealmente independientes, existen 
+\begin_inset Formula $f_{1},\dots,f_{n}\in X^{*}$
+\end_inset
+
+ con cada 
+\begin_inset Formula $f_{i}(x_{j})=\delta_{ij}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Todo subespacio de 
+\begin_inset Formula $X$
+\end_inset
+
+ de dimensión finita posee un complementario topológico.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $Y\leq X$
+\end_inset
+
+, la 
+\series bold
+restricción
+\series default
+ 
+\begin_inset Formula $\psi:X^{*}\to Y^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $f\mapsto f|_{Y}$
+\end_inset
+
+, es lineal, continua, suprayectiva y abierta.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $Y\leq X$
+\end_inset
+
+ y 
+\begin_inset Formula $X^{*}$
+\end_inset
+
+ es separable, 
+\begin_inset Formula $Y^{*}$
+\end_inset
+
+ también.
+\end_layout
+
+\begin_layout Subsection
+Versión geométrica
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sean 
+\begin_inset Formula $E$
+\end_inset
+
+ un e.l.c.
+ y 
+\begin_inset Formula $F\leq E$
+\end_inset
+
+, toda 
+\begin_inset Formula $u\in F'$
+\end_inset
+
+ se extiende a una 
+\begin_inset Formula $f\in E'$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c.:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $F\leq E$
+\end_inset
+
+, la restricción 
+\begin_inset Formula $E'\to F'$
+\end_inset
+
+, 
+\begin_inset Formula $f\mapsto f|_{F}$
+\end_inset
+
+, es suprayectiva.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $x\in E\setminus0$
+\end_inset
+
+ existe 
+\begin_inset Formula $f\in E'$
+\end_inset
+
+ con 
+\begin_inset Formula $f(x)\neq0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\{x_{1},\dots,x_{n}\}\subseteq E$
+\end_inset
+
+ linealmente independiente, existen 
+\begin_inset Formula $f_{1},\dots,f_{n}\in E'$
+\end_inset
+
+ con cada 
+\begin_inset Formula $f_{i}(x_{j})=\delta_{ij}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+-e.v.t., 
+\begin_inset Formula $f\in E'\setminus0$
+\end_inset
+
+ y 
+\begin_inset Formula $A\subseteq E$
+\end_inset
+
+ es un abierto convexo no vacío, 
+\begin_inset Formula $f(A)\subseteq\mathbb{R}$
+\end_inset
+
+ es un intervalo abierto.
+ 
+\series bold
+Demostración:
+\series default
+ Si fuera 
+\begin_inset Formula $f(A)=\{p\}$
+\end_inset
+
+ para cierto 
+\begin_inset Formula $p\in\mathbb{R}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $A\subseteq\ker(f-p)$
+\end_inset
+
+, pero como 
+\begin_inset Formula $f\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $\ker(f-p)<E$
+\end_inset
+
+ y por tanto tiene interior vacío, luego 
+\begin_inset Formula $A=\emptyset\#$
+\end_inset
+
+.
+ Para ver que es un intervalo, sean 
+\begin_inset Formula $x,y\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $f(x)<f(y)$
+\end_inset
+
+, por convexidad, si 
+\begin_inset Formula $\psi:\mathbb{R}\to E$
+\end_inset
+
+ viene dada por 
+\begin_inset Formula $\psi(t)\coloneqq(1-t)x+ty$
+\end_inset
+
+, 
+\begin_inset Formula $\psi([0,1])\subseteq A$
+\end_inset
+
+, pero 
+\begin_inset Formula $\psi$
+\end_inset
+
+ es continua y por tanto también lo es 
+\begin_inset Formula $f\circ\psi:\mathbb{R}\to\mathbb{R}$
+\end_inset
+
+, y para 
+\begin_inset Formula $z\in[f(x),f(y)]$
+\end_inset
+
+, por el teorema de Bolzano existe 
+\begin_inset Formula $t\in[0,1]$
+\end_inset
+
+ con 
+\begin_inset Formula $z=f(\psi(t))\in f(A)$
+\end_inset
+
+.
+ Ahora bien, como 
+\begin_inset Formula $A$
+\end_inset
+
+ es abierto, 
+\begin_inset Formula $A-x$
+\end_inset
+
+ es entorno del 0 y por tanto absorbente, y dada la función lineal 
+\begin_inset Formula $\phi(t)\coloneqq\psi(t)-x=t(y-x)$
+\end_inset
+
+, existe 
+\begin_inset Formula $\rho_{0}>0$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $\rho>\rho_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $\phi(-1)\in\rho(A-x)$
+\end_inset
+
+, luego 
+\begin_inset Formula $\phi(-\frac{1}{\rho}),\phi(\frac{2}{\rho})\in A-x$
+\end_inset
+
+ y 
+\begin_inset Formula $\psi((-\frac{1}{\rho},1))=\phi((-\frac{1}{\rho},1))+x\subseteq A$
+\end_inset
+
+ y, como 
+\begin_inset Formula $f\circ\psi:\mathbb{R}\to\mathbb{R}$
+\end_inset
+
+ es afín no degenerada y por tanto un homeomorfismo, 
+\begin_inset Formula $f(\psi((-\frac{1}{\rho},1)))\subseteq f(A)$
+\end_inset
+
+ es un entorno abierto de 
+\begin_inset Formula $x$
+\end_inset
+
+, pero análogamente hay un entorno abierto de 
+\begin_inset Formula $y$
+\end_inset
+
+, y como 
+\begin_inset Formula $f(A)$
+\end_inset
+
+ tiene al menos dos puntos distintos, queda que 
+\begin_inset Formula $f(A)$
+\end_inset
+
+ es abierta.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\psi([0,1])\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $\psi|_{[0,1]}:[0,1]\to\psi([0,1])$
+\end_inset
+
+ es un homeomorfismo, luego 
+\begin_inset Formula $f\circ\psi:[0,1]\to\mathbb{R}$
+\end_inset
+
+ es continua y, para 
+\begin_inset Formula $z\in(f(x),f(y))$
+\end_inset
+
+, por el teorema de Bolzano existe 
+\begin_inset Formula $t\in[0,1]$
+\end_inset
+
+ con 
+\begin_inset Formula $z=f(\psi(t))\in f(A)$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $A$
+\end_inset
+
+ es abierto, 
+\begin_inset Formula $A-x$
+\end_inset
+
+ es un entorno del 0, luego es absorbente y existe 
+\begin_inset Formula $\rho_{0}>0$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $\rho>\rho_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $\psi(-1)\in\rho(A-x)$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\psi(-\frac{1}{\rho})\in A$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $(-\frac{1}{\rho_{0}},1)\subseteq A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un espacio vectorial, un 
+\series bold
+hiperplano
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+ es un subespacio propio de 
+\begin_inset Formula $E$
+\end_inset
+
+ y una 
+\series bold
+variedad afín
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+ es un conjunto 
+\begin_inset Formula $x_{0}+F$
+\end_inset
+
+ con 
+\begin_inset Formula $x_{0}\in E$
+\end_inset
+
+ y 
+\begin_inset Formula $F\leq E$
+\end_inset
+
+, que se llama 
+\series bold
+hiperplano afín
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+ si 
+\begin_inset Formula $F$
+\end_inset
+
+ es un hiperplano de 
+\begin_inset Formula $E$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.v.t., 
+\begin_inset Formula $M\subseteq E$
+\end_inset
+
+ es un hiperplano afín si y sólo si existen 
+\begin_inset Formula $f:E\to\mathbb{K}$
+\end_inset
+
+ lineal y 
+\begin_inset Formula $a\in\mathbb{K}$
+\end_inset
+
+ con 
+\begin_inset Formula $M=\{x\in X\mid f(x)=a\}$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $M$
+\end_inset
+
+ es cerrado si y sólo si 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Mazur:
+\series default
+ Sean 
+\begin_inset Formula $E$
+\end_inset
+
+ un e.v.t., 
+\begin_inset Formula $M\subseteq E$
+\end_inset
+
+ una variedad afín y 
+\begin_inset Formula $A\subseteq E$
+\end_inset
+
+ un abierto convexo no vacío disjunto de 
+\begin_inset Formula $M$
+\end_inset
+
+, existe un hiperplano afín cerrado de 
+\begin_inset Formula $E$
+\end_inset
+
+ disjunto de 
+\begin_inset Formula $A$
+\end_inset
+
+ que contiene a 
+\begin_inset Formula $M$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Podemos suponer por traslación que 
+\begin_inset Formula $0\in A$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $A$
+\end_inset
+
+ es absorbente y tiene asociado un funcional de Minkowski 
+\begin_inset Formula $p$
+\end_inset
+
+ tal que 
+\begin_inset Formula $A=\{x\in E\mid p(x)<1\}$
+\end_inset
+
+ y, como 
+\begin_inset Formula $A$
+\end_inset
+
+ es abierto, 
+\begin_inset Formula $p$
+\end_inset
+
+ es continua.
+ Sean entonces 
+\begin_inset Formula $x_{0}\in E$
+\end_inset
+
+ y 
+\begin_inset Formula $F\leq E$
+\end_inset
+
+ con 
+\begin_inset Formula $M=x_{0}+F$
+\end_inset
+
+, 
+\begin_inset Formula $x_{0}\notin F$
+\end_inset
+
+ ya que de serlo sería 
+\begin_inset Formula $M=F\ni0$
+\end_inset
+
+, luego 
+\begin_inset Formula $F\cap\text{span}\{x_{0}\}=0$
+\end_inset
+
+ y podemos definir 
+\begin_inset Formula $u:F\oplus\text{span}\{x_{0}\}\to\mathbb{K}$
+\end_inset
+
+ como 
+\begin_inset Formula $u(y+\lambda x_{0})\coloneqq\lambda$
+\end_inset
+
+ para 
+\begin_inset Formula $y\in F$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\in\mathbb{K}$
+\end_inset
+
+, que es lineal.
+ Ahora bien, para 
+\begin_inset Formula $\lambda\neq0$
+\end_inset
+
+ es 
+\begin_inset Formula $|u(y+\lambda x_{0})|=|\lambda|\leq|\lambda|p(\tfrac{y}{\lambda}+x_{0})\leq p(y+\lambda x_{0})$
+\end_inset
+
+, donde en la primera desigualdad usamos que 
+\begin_inset Formula $\frac{y}{\lambda}+x_{0}\in M\subseteq A^{\complement}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $p(\frac{y}{\lambda}+x_{0})\geq1$
+\end_inset
+
+, y para 
+\begin_inset Formula $\lambda=0$
+\end_inset
+
+, 
+\begin_inset Formula $|u(y)|=0\leq p(y)$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $|u|\leq p|_{F\oplus\text{span}\{x_{0}\}}$
+\end_inset
+
+ y, por el teorema de Sobczyk, 
+\begin_inset Formula $u$
+\end_inset
+
+ se extiende a una 
+\begin_inset Formula $f:E\to\mathbb{K}$
+\end_inset
+
+ lineal con 
+\begin_inset Formula $|f|\leq p$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua y, si 
+\begin_inset Formula $H\coloneqq\{x\in E\mid f(x)=1\}$
+\end_inset
+
+, 
+\begin_inset Formula $f(E)=1$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $E\subseteq H$
+\end_inset
+
+ y, para 
+\begin_inset Formula $x\in H$
+\end_inset
+
+, 
+\begin_inset Formula $f(x)=1\leq p(x)$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $x\notin A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $E$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+-e.v.t., 
+\begin_inset Formula $f\in E'$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in\mathbb{R}$
+\end_inset
+
+, llamamos 
+\series bold
+semiespacios abiertos
+\series default
+ determinados por 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\{x\in E\mid f(x)<\alpha\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{x\in E\mid f(x)>\alpha\}$
+\end_inset
+
+, y 
+\series bold
+semiespacios cerrados
+\series default
+ determinados por 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\{x\in E\mid f(x)\leq\alpha\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{x\in E\mid f(x)\geq\alpha\}$
+\end_inset
+
+, y 
+\begin_inset Formula $H\coloneqq\{x\in E\mid f(x)=\alpha\}$
+\end_inset
+
+ 
+\series bold
+separa
+\series default
+ 
+\begin_inset Formula $A,B\subseteq E$
+\end_inset
+
+ si cada uno está en un semiespacio cerrado distinto de 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ están 
+\series bold
+separados
+\series default
+, y 
+\series bold
+separa estrictamente
+\series default
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ si cada uno está en un semiespacio abierto distinto de 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ están estrictamente separados.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teoremas de separación:
+\end_layout
+
+\begin_layout Enumerate
+En un 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+-e.v.t.
+ todo par de abiertos convexos disjuntos no vacíos está separado.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $E$
+\end_inset
+
+ el 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+-e.v.t.
+ y 
+\begin_inset Formula $A,B\subseteq E$
+\end_inset
+
+ tales conjuntos, 
+\begin_inset Formula $A-B$
+\end_inset
+
+ es un abierto no vacío que no contiene al 0, con lo que el teorema de Mizur
+ nos da un hiperplano cerrado 
+\begin_inset Formula $H=\{x\in E\mid f(x)=\beta\}$
+\end_inset
+
+, con 
+\begin_inset Formula $f\in E'$
+\end_inset
+
+ y 
+\begin_inset Formula $\beta\in\mathbb{R}$
+\end_inset
+
+, que contiene al 0 y es disjunto de 
+\begin_inset Formula $A-B$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f(A-B)\subseteq\mathbb{R}$
+\end_inset
+
+ es convexo.
+ Como 
+\begin_inset Formula $\beta=f(0)=0$
+\end_inset
+
+, 
+\begin_inset Formula $0\notin f(A-B)$
+\end_inset
+
+, pero 
+\begin_inset Formula $f(A-B)$
+\end_inset
+
+ es un intervalo, luego 
+\begin_inset Formula $f(A-B)\subseteq\mathbb{R}^{+}$
+\end_inset
+
+ o 
+\begin_inset Formula $f(A-B)\subseteq\mathbb{R}^{-}$
+\end_inset
+
+.
+ Si, por ejemplo, 
+\begin_inset Formula $f(A-B)\subseteq\mathbb{R}^{-}$
+\end_inset
+
+, para 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, 
+\begin_inset Formula $f(a)<f(b)$
+\end_inset
+
+, luego existe 
+\begin_inset Formula $\alpha\in[\sup_{a\in A}f(a),\inf_{b\in B}f(b)]$
+\end_inset
+
+, y como 
+\begin_inset Formula $f(A)$
+\end_inset
+
+ y 
+\begin_inset Formula $f(B)$
+\end_inset
+
+ son intervalos abiertos, para 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, 
+\begin_inset Formula $f(a)<\alpha<f(b)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+-e.l.c.
+ y 
+\begin_inset Formula $K,F\subseteq E$
+\end_inset
+
+ son convexos disjuntos no vacíos con 
+\begin_inset Formula $K$
+\end_inset
+
+ compacto y 
+\begin_inset Formula $F$
+\end_inset
+
+ cerrado, existen 
+\begin_inset Formula $f\in E'$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha\in\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ con 
+\begin_inset Formula $f(y)\leq\alpha-\varepsilon<\alpha<f(z)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $y\in K$
+\end_inset
+
+ y 
+\begin_inset Formula $z\in F$
+\end_inset
+
+ y tales que 
+\begin_inset Formula $f|_{K}$
+\end_inset
+
+ alcanza 
+\begin_inset Formula $\alpha-\varepsilon$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $K$
+\end_inset
+
+ y 
+\begin_inset Formula $F$
+\end_inset
+
+ están estrictamente separados.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Como 
+\begin_inset Formula $K-F$
+\end_inset
+
+ es cerrado y no contiene al 0, 
+\begin_inset Formula $E\setminus(K-F)\in{\cal E}(0)$
+\end_inset
+
+, luego existe 
+\begin_inset Formula $W\in{\cal E}(0)$
+\end_inset
+
+ con 
+\begin_inset Formula $W+W\subseteq E\setminus(K-F)$
+\end_inset
+
+ que podemos tomar absolutamente conexo y, si 
+\begin_inset Formula $k\in K$
+\end_inset
+
+, 
+\begin_inset Formula $f\in F$
+\end_inset
+
+ y 
+\begin_inset Formula $u,v\in W$
+\end_inset
+
+, 
+\begin_inset Formula $k-f\in K-F$
+\end_inset
+
+ y 
+\begin_inset Formula $u-v\in W+W\subseteq E\setminus(K-F)$
+\end_inset
+
+, luego 
+\begin_inset Formula $k-f\neq u-v$
+\end_inset
+
+ y 
+\begin_inset Formula $k+v\neq f+u$
+\end_inset
+
+, y 
+\begin_inset Formula $K+W$
+\end_inset
+
+ y 
+\begin_inset Formula $F+W$
+\end_inset
+
+ son abiertos disjuntos.
+ Es fácil ver que la suma de conexos es conexa, luego 
+\begin_inset Formula $K+W$
+\end_inset
+
+ y 
+\begin_inset Formula $F+W$
+\end_inset
+
+ son conexos y, por el primer teorema de separación, existen 
+\begin_inset Formula $f\in E'$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $f(k)<\alpha<f(z)$
+\end_inset
+
+ para 
+\begin_inset Formula $k\in K$
+\end_inset
+
+ y 
+\begin_inset Formula $z\in F$
+\end_inset
+
+, pero como 
+\begin_inset Formula $f(K)$
+\end_inset
+
+ es compacto, 
+\begin_inset Formula $\max f(K)<\alpha$
+\end_inset
+
+ y basta tomar 
+\begin_inset Formula $\varepsilon\coloneqq\alpha-\max f(K)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Con esto, si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-e.l.c.
+ y 
+\begin_inset Formula $K,F\subseteq E$
+\end_inset
+
+ son convexos disjuntos, 
+\begin_inset Formula $A$
+\end_inset
+
+ es compacto, 
+\begin_inset Formula $B$
+\end_inset
+
+ es cerrado y uno de los dos es absolutamente convexo, existe 
+\begin_inset Formula $u\in E'$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\sup_{x\in A}|u(x)|<\inf_{y\in B}|u(y)|$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c.
+ y 
+\begin_inset Formula $A\subseteq E$
+\end_inset
+
+, 
+\begin_inset Formula $\overline{\text{co}(A)}$
+\end_inset
+
+ es la intersección de todos los semiespacios cerrados de 
+\begin_inset Formula $E$
+\end_inset
+
+ que contienen a 
+\begin_inset Formula $A$
+\end_inset
+
+, y en particular todo conjunto convexo y cerrado es la intersección de
+ los semiespacios cerrados que lo contienen.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un espacio vectorial con topologías 
+\begin_inset Formula ${\cal S}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+ localmente convexas Hausdorff y 
+\begin_inset Formula $(E,{\cal S})'=(E,{\cal T})'$
+\end_inset
+
+, 
+\begin_inset Formula $(E,{\cal S})$
+\end_inset
+
+ y 
+\begin_inset Formula $(E,{\cal T})$
+\end_inset
+
+ tienen los mismos convexos cerrados.
+ Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c.
+ entonces 
+\begin_inset Formula $(E,{\cal T})'=(E,\sigma(E,E'))'$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c.:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $F\leq E$
+\end_inset
+
+, 
+\begin_inset Formula $\overline{F}=\{x\in E\mid\forall f\in E',(f|_{F}=0\implies f(x)=0)\}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $S\subseteq E$
+\end_inset
+
+ es total si y sólo si 
+\begin_inset Formula $\{f\in E'\mid f|_{S}=0\}=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Normas convexas
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+bidual
+\series default
+ del espacio normado 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ al dual del dual de 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $X^{**}$
+\end_inset
+
+, con la norma dual, que es un espacio de Banach.
+\end_layout
+
+\begin_layout Standard
+La función 
+\begin_inset Formula $\hat{}:X\to X^{**}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\hat{x}(f)\coloneqq f(x)$
+\end_inset
+
+ es una isometría, con lo que 
+\begin_inset Formula $\overline{\text{Im}\hat{}}$
+\end_inset
+
+ es un modelo para la compleción de 
+\begin_inset Formula $X$
+\end_inset
+
+ identificando cada 
+\begin_inset Formula $x$
+\end_inset
+
+ con 
+\begin_inset Formula $\hat{x}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $Y\leq X$
+\end_inset
+
+ cerrado, 
+\begin_inset Formula $Q:X\to\frac{X}{Y}$
+\end_inset
+
+ la aplicación cociente e 
+\begin_inset Formula $Y'\coloneqq\{f\in X^{*}\mid f(Y)=0\}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\alpha:\frac{X^{*}}{Y'}\to Y^{*}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\alpha(\overline{f})\coloneqq f|_{Y}$
+\end_inset
+
+ es un isomorfismo isométrico.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\beta:\left(\frac{X}{Y}\right)^{*}\to Y'$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\beta(\overline{g})\coloneqq g\circ Q$
+\end_inset
+
+ es un isomorfismo isométrico.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una norma 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ es 
+\series bold
+estrictamente convexa
+\series default
+ si 
+\begin_inset Formula $\forall x,y\in S_{X},\left(x\neq y\implies\left\Vert \frac{x+y}{2}\right\Vert <1\right)$
+\end_inset
+
+.
+ 
+\series bold
+Teorema de Taylor-Foguel:
+\series default
+ Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio normado, 
+\begin_inset Formula $X^{*}$
+\end_inset
+
+ es estrictamente convexo si y sólo si para 
+\begin_inset Formula $Y\leq X$
+\end_inset
+
+ e 
+\begin_inset Formula $f\in Y^{*}$
+\end_inset
+
+ existe una única extensión 
+\begin_inset Formula $\hat{f}\in X^{*}$
+\end_inset
+
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert\hat{f}\Vert=\Vert f\Vert$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $p\in(1,\infty)$
+\end_inset
+
+ y 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, en 
+\begin_inset Formula $(\mathbb{K}^{n},\Vert\cdot\Vert_{p})$
+\end_inset
+
+ y 
+\begin_inset Formula $(\ell^{p},\Vert\cdot\Vert_{p})$
+\end_inset
+
+ las normas duales son estrictamente convexas, mientras que esto no ocurre
+ cuando 
+\begin_inset Formula $p=1$
+\end_inset
+
+ o 
+\begin_inset Formula $p=\infty$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Las extensiones de Hann-Banach pueden ser infinitas; por ejemplo, si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es el subespacio de 
+\begin_inset Formula $({\cal C}([0,1]),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ de las funciones constantes y 
+\begin_inset Formula $g\in Y^{*}$
+\end_inset
+
+ viene dada por 
+\begin_inset Formula $g(y)\coloneqq y(0)$
+\end_inset
+
+, para 
+\begin_inset Formula $t\in[0,1]$
+\end_inset
+
+, 
+\begin_inset Formula $f_{t}\in X^{*}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f_{t}(x)\coloneqq x(t)$
+\end_inset
+
+ es una extensión lineal de 
+\begin_inset Formula $g$
+\end_inset
+
+ que conserva la norma.
+\end_layout
+
+\begin_layout Subsection
+Límites de Banach
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $c$
+\end_inset
+
+ es el espacio de las sucesiones convergentes, existe 
+\begin_inset Formula $L\in(\ell^{\infty})^{*}$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert L\Vert=1$
+\end_inset
+
+ y 
+\begin_inset Formula $L(x)=\lim_{n}x_{n}$
+\end_inset
+
+ para 
+\begin_inset Formula $x\in c$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, 
+\begin_inset Formula $L(x)=L((x_{2},x_{3},\dots,x_{n},\dots))$
+\end_inset
+
+ y, si cada 
+\begin_inset Formula $x_{n}\geq0$
+\end_inset
+
+, 
+\begin_inset Formula $L(x)\geq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Espacios vectoriales ordenados
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+espacio vectorial ordenado
+\series default
+ es un conjunto preordenado 
+\begin_inset Formula $(X,\apprle)$
+\end_inset
+
+ donde 
+\begin_inset Formula $X$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+-espacio vectorial y, para 
+\begin_inset Formula $\alpha\in\mathbb{R}^{\geq0}$
+\end_inset
+
+ y 
+\begin_inset Formula $x,y,z\in X$
+\end_inset
+
+ con 
+\begin_inset Formula $x\leq y$
+\end_inset
+
+, 
+\begin_inset Formula $x+z\leq y+z$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha x\leq\alpha y$
+\end_inset
+
+.
+ Un 
+\series bold
+cono
+\series default
+ en un 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $X$
+\end_inset
+
+ es un 
+\begin_inset Formula $P\subseteq X$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $\alpha\in\mathbb{R}^{\geq0}$
+\end_inset
+
+ y 
+\begin_inset Formula $x,y\in P$
+\end_inset
+
+, 
+\begin_inset Formula $x+y\in P$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha x\in P$
+\end_inset
+
+ y 
+\begin_inset Formula $P\cap(-P)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $(X,\apprle)$
+\end_inset
+
+ es un espacio vectorial ordenado, 
+\begin_inset Formula $\{x\in X\mid x\geq0\}$
+\end_inset
+
+ es un cono si y sólo si 
+\begin_inset Formula $\apprle$
+\end_inset
+
+ es antisimétrica, y si 
+\begin_inset Formula $P\subseteq_{\mathbb{R}}X$
+\end_inset
+
+ es un cono, 
+\begin_inset Formula $x\leq y\iff y-x\in P$
+\end_inset
+
+ define un orden parcial en 
+\begin_inset Formula $P$
+\end_inset
+
+ tal que 
+\begin_inset Formula $(X,\leq)$
+\end_inset
+
+ es un espacio vectorial ordenado.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $(X,\apprle)$
+\end_inset
+
+ es un espacio vectorial ordenado, 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es 
+\series bold
+cofinal
+\series default
+ si 
+\begin_inset Formula $\forall x\geq0,\exists a\in A:a\apprge x$
+\end_inset
+
+, y 
+\begin_inset Formula $e\in X$
+\end_inset
+
+ es 
+\series bold
+unidad de orden
+\series default
+ si 
+\begin_inset Formula $\forall x\in X,\exists n\in\mathbb{N}:-ne\apprle x\apprle ne$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $\{ne\}_{n\in\mathbb{N}}$
+\end_inset
+
+ es cofinal.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K$
+\end_inset
+
+ es un espacio compacto, en el espacio vectorial ordenado 
+\begin_inset Formula $(C(K),\leq)$
+\end_inset
+
+ de funciones continuas 
+\begin_inset Formula $K\to\mathbb{R}$
+\end_inset
+
+ con el orden 
+\begin_inset Formula $f\leq g\iff\forall x\in K,f(x)\leq f(x)$
+\end_inset
+
+, todas las funciones que no se anulan son unidades de orden.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(C(\mathbb{R}),\leq)$
+\end_inset
+
+ no tiene unidades de orden.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $(X,\apprle)$
+\end_inset
+
+ e 
+\begin_inset Formula $(Y,\lessapprox)$
+\end_inset
+
+ son espacios vectoriales ordenados, 
+\begin_inset Formula $T:X\to Y$
+\end_inset
+
+ lineal es 
+\series bold
+positiva
+\series default
+ si 
+\begin_inset Formula $\forall x\apprge0,Tx\gtrapprox0$
+\end_inset
+
+.
+ Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $(X,\apprle)$
+\end_inset
+
+ es un espacio vectorial ordenado, 
+\begin_inset Formula $Y\leq X$
+\end_inset
+
+ cofinal y 
+\begin_inset Formula $f:Y\to\mathbb{R}$
+\end_inset
+
+ lineal positiva, 
+\begin_inset Formula $f$
+\end_inset
+
+ se extiende a una 
+\begin_inset Formula $\hat{f}:X\to\mathbb{R}$
+\end_inset
+
+ lineal positiva.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Con esto, si 
+\begin_inset Formula $e$
+\end_inset
+
+ es una unidad de orden de 
+\begin_inset Formula $(X,\leq)$
+\end_inset
+
+ e 
+\begin_inset Formula $Y\leq X$
+\end_inset
+
+ con 
+\begin_inset Formula $e\in Y$
+\end_inset
+
+, toda función lineal positiva 
+\begin_inset Formula $Y\to\mathbb{R}$
+\end_inset
+
+ se extiende a una función 
+\begin_inset Formula $X\to\mathbb{R}$
+\end_inset
+
+ lineal positiva.
+\end_layout
+
+\begin_layout Section
+Propiedad de extensión
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Helly:
+\series default
+ En 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+, la intersección de 
+\begin_inset Formula $m>n$
+\end_inset
+
+ conjuntos convexos es no vacía si y sólo si la intersección de cada 
+\begin_inset Formula $n+1$
+\end_inset
+
+ de ellos es no vacía.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio normado 
+\begin_inset Formula $X$
+\end_inset
+
+ y familias 
+\begin_inset Formula $\{x_{i}\}_{i\in I}\subseteq X$
+\end_inset
+
+ y 
+\begin_inset Formula $\{r_{i}\}_{i\in I}\subseteq\mathbb{R}^{+}$
+\end_inset
+
+, la familia de bolas cerradas 
+\begin_inset Formula $(\overline{B(x_{i},r_{i})})_{i\in I}$
+\end_inset
+
+ tiene la 
+\series bold
+propiedad de intersección débil
+\series default
+ si 
+\begin_inset Formula $\forall f\in B_{X^{*}},\bigcap_{i\in I}B(f(x_{i}),r_{i})\neq\emptyset$
+\end_inset
+
+, si y sólo si para 
+\begin_inset Formula $J\subseteq I$
+\end_inset
+
+ finito y 
+\begin_inset Formula $\{a_{j}\}_{j\in J}\subseteq\mathbb{K}$
+\end_inset
+
+ con 
+\begin_inset Formula $\sum_{j\in J}a_{j}=0$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+\left\Vert \sum_{j\in J}a_{j}x_{j}\right\Vert \leq\sum_{j\in J}|a_{j}|r_{j}.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\mathbb{K}=\mathbb{R}$
+\end_inset
+
+, esto equivale a que las bolas se corten dos a dos.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\mathbb{K}=\mathbb{C}$
+\end_inset
+
+, la segunda definición se puede restringir sólo a los 
+\begin_inset Formula $J\subseteq I$
+\end_inset
+
+ con 
+\begin_inset Formula $|J|=3$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio normado 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ cumple:
+\end_layout
+
+\begin_layout Enumerate
+La 
+\series bold
+propiedad de extensión
+\series default
+, si para cada 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio normado 
+\begin_inset Formula $Y$
+\end_inset
+
+, 
+\begin_inset Formula $Y_{0}\leq Y$
+\end_inset
+
+ y 
+\begin_inset Formula $T_{0}\in{\cal L}(Y_{0},X)$
+\end_inset
+
+, 
+\begin_inset Formula $T_{0}$
+\end_inset
+
+ se extiende a una 
+\begin_inset Formula $T\in{\cal L}(Y,X)$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert T\Vert=\Vert T_{0}\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La 
+\series bold
+propiedad de extensión 
+\begin_inset Quotes cld
+\end_inset
+
+inmediata
+\begin_inset Quotes crd
+\end_inset
+
+
+\series default
+, si cumple la de extensión pero considerando sólo el caso en que 
+\begin_inset Formula $Y_{0}$
+\end_inset
+
+ es de codimensión 1 en 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La 
+\series bold
+propiedad de intersección
+\series default
+ si toda familia de bolas cerradas de 
+\begin_inset Formula $X$
+\end_inset
+
+ que cumple la propiedad de intersección débil tiene intersección no vacía.
+\end_layout
+
+\begin_layout Enumerate
+La 
+\series bold
+propiedad de intersección binaria
+\series default
+ si toda familia de bolas cerradas de 
+\begin_inset Formula $X$
+\end_inset
+
+ que se cortan dos a dos tiene intersección no vacía.
+\end_layout
+
+\begin_layout Enumerate
+La 
+\series bold
+propiedad de proyección
+\series default
+ si para todo 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio normado que contiene a 
+\begin_inset Formula $X$
+\end_inset
+
+ como subespacio existe 
+\begin_inset Formula $P\in{\cal L}(Y,X)$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert P\Vert=1$
+\end_inset
+
+ suprayectiva e idempotente, que llamamos una 
+\series bold
+proyección
+\series default
+ de 
+\begin_inset Formula $Y$
+\end_inset
+
+ sobre 
+\begin_inset Formula $X$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $X$
+\end_inset
+
+ cumple la propiedad de extensión si y sólo si cumple la propiedad de extensión
+ inmediata, en cuyo caso 
+\begin_inset Formula $X$
+\end_inset
+
+ es de Banach.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un espacio compacto 
+\begin_inset Formula $K$
+\end_inset
+
+ es 
+\series bold
+stoniano
+\series default
+ si la clausura de cada abierto de 
+\begin_inset Formula $K$
+\end_inset
+
+ es un abierto.
+ 
+\end_layout
+
+\begin_layout Standard
+Dado un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Banach 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Las propiedades de extensión, intersección y proyección son equivalentes.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Teorema de Nachbin-Goodner-Kelly-Hasumi:
+\series default
+ Estas propiedades equivalen a que 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ sea isométricamente isomorfo a 
+\begin_inset Formula $({\cal C}(K,\mathbb{K}),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ para algún compacto stoniano 
+\begin_inset Formula $K$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\mathbb{K}=\mathbb{R}$
+\end_inset
+
+, estas equivalen a la propiedad de intersección binaria.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\mathbb{K}=\mathbb{C}$
+\end_inset
+
+, estas equivalen a la propiedad de intersección pero limitando las subfamilias
+ de las familias de bolas a que sean de cardinal 3.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Teorema de la acotación uniforme
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $X$
+\end_inset
+
+ un espacio topológico, 
+\begin_inset Formula $S\subseteq X$
+\end_inset
+
+ es 
+\series bold
+denso en ninguna parte
+\series default
+ o 
+\series bold
+raro
+\series default
+ si su clausura tiene interior vacío, 
+\begin_inset Formula $\mathring{\overline{S}}=\emptyset$
+\end_inset
+
+, 
+\series bold
+de primera categoría
+\series default
+ si es unión numerable de conjuntos raros, 
+\series bold
+de segunda categoría
+\series default
+ en otro caso y 
+\series bold
+
+\begin_inset Formula $G_{\delta}$
+\end_inset
+
+
+\series default
+ si es intersección numerable de abiertos.
+ 
+\begin_inset Formula $T$
+\end_inset
+
+ es de segunda categoría en sí mismo si y sólo si la intersección numerable
+ de abiertos densos en no vacía.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un espacio topológico es 
+\series bold
+de Baire
+\series default
+ si la intersección numerable de abiertos densos es densa, en cuyo caso
+ es de segunda categoría en sí mismo.
+ 
+\series bold
+Teorema de Baire:
+\series default
+ Todo espacio métrico completo es de Baire.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+ un espacio métrico, 
+\begin_inset Formula $(G_{n})_{n}$
+\end_inset
+
+ una sucesión de abiertos densos y 
+\begin_inset Formula $V\subseteq M$
+\end_inset
+
+ abierto arbitrario, queremos definir una sucesión de bolas 
+\begin_inset Formula $(\overline{B(x_{n},r_{n})})_{n}$
+\end_inset
+
+ cada una contenida en 
+\begin_inset Formula $V\cap G_{n}\cap\overline{B(x_{n-1},r_{n-1})}$
+\end_inset
+
+ y con 
+\begin_inset Formula $r_{n}<\frac{1}{2^{n}}$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $G_{0}$
+\end_inset
+
+ es denso, 
+\begin_inset Formula $V\cap G_{0}\neq\emptyset$
+\end_inset
+
+ y existen 
+\begin_inset Formula $x_{0}\in M$
+\end_inset
+
+ y 
+\begin_inset Formula $r_{0}\in(0,1)$
+\end_inset
+
+ con 
+\begin_inset Formula $\overline{B(x_{0},r_{0})}\subseteq V\cap G_{0}$
+\end_inset
+
+, y para 
+\begin_inset Formula $n>0$
+\end_inset
+
+, como 
+\begin_inset Formula $G_{n}$
+\end_inset
+
+ es denso, por inducción existen 
+\begin_inset Formula $x_{n}\in M$
+\end_inset
+
+ y 
+\begin_inset Formula $r_{n}\in(0,\frac{1}{2^{n}})$
+\end_inset
+
+ con 
+\begin_inset Formula $\overline{B(x_{n},r_{n})}\subseteq V\cap B(x_{n-1},r_{n-1})\cap G_{n}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $(x_{n})_{n}$
+\end_inset
+
+ es de Cauchy por ser 
+\begin_inset Formula $x_{m}\in B(x_{n},r_{n})$
+\end_inset
+
+ para 
+\begin_inset Formula $m\geq n$
+\end_inset
+
+ y 
+\begin_inset Formula $\lim_{n}r_{n}=0$
+\end_inset
+
+, luego existe 
+\begin_inset Formula $L\coloneqq\lim_{n}x_{n}\in V\cap\bigcap_{n}G_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\bigcap_{n}G_{n}$
+\end_inset
+
+ es denso.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+ no es completo esto no se cumple; por ejemplo, en 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ con la métrica inducida por la de 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+, para cada 
+\begin_inset Formula $q\in\mathbb{Q}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{Q}\setminus\{q\}$
+\end_inset
+
+ es denso, pero la intersección numerable 
+\begin_inset Formula $\bigcap_{q\in\mathbb{Q}}\mathbb{Q}\setminus\{q\}=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Con esto, si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de Banach, su dimensión algebraica es finita o no numerable.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de la acotación uniforme:
+\series default
+ Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios normados, 
+\begin_inset Formula $\{A_{i}\}_{i\in I}\subseteq{\cal L}(X,Y)$
+\end_inset
+
+ y 
+\begin_inset Formula $B\coloneqq\{x\in X\mid\sup_{i\in I}\Vert A_{i}(x)\Vert<\infty\}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $B$
+\end_inset
+
+ es de segunda categoría, 
+\begin_inset Formula $\sup_{i\in I}\Vert A_{i}\Vert<\infty$
+\end_inset
+
+ y 
+\begin_inset Formula $B=\emptyset$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de Banach, bien 
+\begin_inset Formula $\sup_{i\in I}\Vert A_{i}\Vert<\infty$
+\end_inset
+
+ o 
+\begin_inset Formula $B^{\complement}$
+\end_inset
+
+ es 
+\begin_inset Formula $G_{\delta}$
+\end_inset
+
+ denso en 
+\begin_inset Formula $X$
+\end_inset
+
+, de modo que o 
+\begin_inset Formula $\sup_{i\in I}\Vert A_{i}\Vert<\infty$
+\end_inset
+
+ o 
+\begin_inset Formula $B$
+\end_inset
+
+ es de primera categoría en 
+\begin_inset Formula $X$
+\end_inset
+
+, pero no ambas.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La completitud es necesaria para la segunda parte del teorema, pues 
+\begin_inset Formula $\{f_{n}\}_{n}\subseteq(c_{00},\Vert\cdot\Vert_{\infty})^{*}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f_{n}(x)\coloneqq\sum_{i=1}^{n}x_{i}$
+\end_inset
+
+ es puntualmente acotada pero cada 
+\begin_inset Formula $\Vert f_{n}\Vert=n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach, 
+\begin_inset Formula $Y$
+\end_inset
+
+ un espacio completo y 
+\begin_inset Formula $\{T_{n}\}_{n}\subseteq{\cal L}(X,Y)$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ existe 
+\begin_inset Formula $T(x)\coloneqq\lim_{n}T_{n}(x)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Teorema de Banach-Steinhaus:
+\series default
+ 
+\begin_inset Formula $T$
+\end_inset
+
+ es lineal y continua con
+\begin_inset Formula 
+\[
+\Vert T\Vert\leq\liminf_{n}\Vert T_{n}\Vert\leq\sup_{n}\Vert T_{n}\Vert<\infty.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Es lineal por serlo el límite.
+ 
+\begin_inset Formula $(T_{n}x)_{n}$
+\end_inset
+
+ es acotada para 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ y, por el teorema de la acotación uniforme, 
+\begin_inset Formula $\sup_{n}\Vert T_{n}\Vert<\infty$
+\end_inset
+
+, y si 
+\begin_inset Formula $x\in B_{X}$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert Tx\Vert=\lim_{n}\Vert T_{n}x\Vert\leq\liminf_{n}\Vert T_{n}\Vert\leq\sup_{n}\Vert T_{n}\Vert$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $(T_{n})_{n}$
+\end_inset
+
+ converge uniformemente a 
+\begin_inset Formula $T$
+\end_inset
+
+ en los subconjuntos compactos de 
+\begin_inset Formula $X$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios normados:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es acotado si y sólo si para 
+\begin_inset Formula $f\in X^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $f(A)$
+\end_inset
+
+ es acotado.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de Banach, 
+\begin_inset Formula $A\subseteq X^{*}$
+\end_inset
+
+ es acotado si y sólo si 
+\begin_inset Formula $\{f(x)\}_{f\in A}$
+\end_inset
+
+ es acotado.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $T:X\to Y$
+\end_inset
+
+ es lineal, 
+\begin_inset Formula $T$
+\end_inset
+
+ es continua si y sólo si 
+\begin_inset Formula $\forall g\in Y^{*},g\circ T\in X^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Funciones holomorfas vectoriales
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
+\end_inset
+
+ abierto y 
+\begin_inset Formula $(_{\mathbb{C}}X,\Vert\cdot\Vert)$
+\end_inset
+
+ de Banach, 
+\begin_inset Formula $f:\Omega\to X$
+\end_inset
+
+ es 
+\series bold
+débilmente holomorfa
+\series default
+ en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si para 
+\begin_inset Formula $g\in X^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $g\circ f:\Omega\to\mathbb{C}$
+\end_inset
+
+ es holomorfa, y es 
+\series bold
+holomorfa
+\series default
+ en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si 
+\begin_inset Formula 
+\[
+\forall a\in\Omega,\exists f'(a)\coloneqq\lim_{z\to a}\frac{f(z)-f(a)}{z-a}.
+\]
+
+\end_inset
+
+
+\series bold
+Teorema de Dunford:
+\series default
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ es holomorfa si y sólo si es débilmente holomorfa.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Liouville:
+\series default
+ Si 
+\begin_inset Formula $(_{\mathbb{C}}X,\Vert\cdot\Vert)$
+\end_inset
+
+ es de Banach y 
+\begin_inset Formula $f:\mathbb{C}\to X$
+\end_inset
+
+ es holomorfa con 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ acotada para cada 
+\begin_inset Formula $g\in X^{*}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es constante.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{FVC}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Toda curva 
+\begin_inset Formula $\gamma:[a,b]\to\mathbb{C}^{*}$
+\end_inset
+
+ tiene argumentos continuos, y si 
+\begin_inset Formula $\theta$
+\end_inset
+
+ y 
+\begin_inset Formula $\theta'$
+\end_inset
+
+ son argumentos continuos de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\theta(b)-\theta(a)=\theta'(b)-\theta'(a)$
+\end_inset
+
+.
+ [...] Sean 
+\begin_inset Formula $\gamma:[a,b]\to\mathbb{C}$
+\end_inset
+
+ una curva, 
+\begin_inset Formula $z\notin\gamma^{*}$
+\end_inset
+
+[
+\begin_inset Formula $\coloneqq\text{Im}\gamma$
+\end_inset
+
+] y 
+\begin_inset Formula $\theta$
+\end_inset
+
+ un argumento de 
+\begin_inset Formula $\gamma-z$
+\end_inset
+
+, llamamos [...] 
+\series bold
+índice
+\series default
+ de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ respecto de 
+\begin_inset Formula $z$
+\end_inset
+
+ a
+\begin_inset Formula 
+\[
+\text{Ind}_{\gamma}(z):=\frac{\theta(b)-\theta(a)}{2\pi}.
+\]
+
+\end_inset
+
+[...] Una 
+\series bold
+cadena
+\series default
+ es una expresión de la forma 
+\begin_inset Formula $\Gamma\coloneqq m_{1}\gamma_{1}+\dots+m_{q}\gamma_{q}$
+\end_inset
+
+ donde los 
+\begin_inset Formula $m_{i}$
+\end_inset
+
+ son enteros y los 
+\begin_inset Formula $\gamma_{i}$
+\end_inset
+
+ son caminos.
+ Llamamos 
+\series bold
+soporte
+\series default
+ de 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ a 
+\begin_inset Formula $\Gamma^{*}\coloneqq\bigcup_{k}\gamma_{k}^{*}$
+\end_inset
+
+ [...].
+ Un 
+\series bold
+ciclo
+\series default
+ es una cadena formada por caminos cerrados, y llamamos 
+\series bold
+índice
+\series default
+ de 
+\begin_inset Formula $z\notin\Gamma^{*}$
+\end_inset
+
+ respecto al ciclo 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{Ind}_{\Gamma}(z)\coloneqq\sum_{k}m_{k}\text{Ind}_{\gamma_{k}}(z)$
+\end_inset
+
+.
+ [...] Dado un abierto 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, un ciclo 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ es 
+\series bold
+nulhomólogo
+\series default
+ respecto de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si 
+\begin_inset Formula $\forall z\in\mathbb{C}\setminus\Omega,\text{Ind}_{\Gamma}(z)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sean 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $_{\mathbb{C}}X$
+\end_inset
+
+ de Banach y 
+\begin_inset Formula $f:\Omega\to X$
+\end_inset
+
+ holomorfa:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Teorema de Cauchy:
+\series default
+ Sea 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ un ciclo 
+\begin_inset Formula $\Omega$
+\end_inset
+
+-nulhomólogo,
+\begin_inset Formula 
+\[
+\int_{\Gamma}f=0.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Fórmula de Cauchy:
+\series default
+ Para 
+\begin_inset Formula $z\in\mathbb{C}\setminus\text{Im}\Gamma$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+f(z)\text{Ind}_{\Gamma}(z)=\frac{1}{2\pi\text{i}}\int_{\Gamma}\frac{f(w)}{w-z}\dif w.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+, existe 
+\begin_inset Formula $ $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+, si 
+\begin_inset Formula $\Gamma:[0,2\pi]\to\mathbb{C}$
+\end_inset
+
+ viene dado por 
+\begin_inset Formula $\Gamma(\theta)=a+\rho\text{e}^{\text{i}\theta}$
+\end_inset
+
+ y, para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+a_{n}\coloneqq\frac{f^{(n)}(a)}{n!}=\frac{1}{2\pi\text{i}}\int_{\Gamma}\frac{f(w)}{(w-a)^{n+1}}\dif w\in X,
+\]
+
+\end_inset
+
+existe 
+\begin_inset Formula $\rho>0$
+\end_inset
+
+ con 
+\begin_inset Formula $\overline{B(a,\rho)}\subseteq\Omega$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f(z)=\sum_{n}a_{n}(z-a)^{n}$
+\end_inset
+
+, y la serie converge uniforme y absolutamente en compactos de 
+\begin_inset Formula $B(a,\rho)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Métodos de sumabilidad
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $A\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+, la sucesión 
+\begin_inset Formula $(x_{m})_{m}$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+ es 
+\series bold
+
+\begin_inset Formula $A$
+\end_inset
+
+-convergente
+\series default
+ a 
+\begin_inset Formula $z\in\mathbb{K}$
+\end_inset
+
+ si para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{m}A_{nm}x_{m}$
+\end_inset
+
+ converge a un cierto 
+\begin_inset Formula $y_{n}$
+\end_inset
+
+ e 
+\begin_inset Formula $(y_{n})_{n}$
+\end_inset
+
+ converge a 
+\begin_inset Formula $z$
+\end_inset
+
+, y 
+\begin_inset Formula $A$
+\end_inset
+
+ es un 
+\series bold
+método de sumabilidad permanente
+\series default
+ si para 
+\begin_inset Formula $\{x_{m}\}_{m}\subseteq\mathbb{K}$
+\end_inset
+
+ convergente, 
+\begin_inset Formula $(\sum_{m}A_{nm}x_{m})_{n}$
+\end_inset
+
+ es convergente y 
+\begin_inset Formula $\lim_{n}y_{n}=\lim_{m}x_{m}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+sucesión de medias de Césaro
+\series default
+ de una sucesión 
+\begin_inset Formula $(x_{n})_{n}$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+\left(\frac{x_{1}+\dots+x_{n}}{n}\right)_{n},
+\]
+
+\end_inset
+
+y 
+\begin_inset Formula $(x_{n})_{n}$
+\end_inset
+
+ es 
+\series bold
+convergente Césaro
+\series default
+ si su sucesión de medias de Césaro converge.
+ Toda sucesión convergente es convergente Césaro, pero el recíproco no se
+ cumple.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, la 
+\series bold
+matriz de Césaro
+\series default
+,
+\begin_inset Formula 
+\[
+\left(\frac{1}{i}\chi_{\{j\leq i\}}\right)_{i,j\geq1}=\begin{pmatrix}1\\
+\frac{1}{2} & \frac{1}{2}\\
+\frac{1}{3} & \frac{1}{3} & \frac{1}{3}\\
+\vdots & \vdots & \vdots & \ddots
+\end{pmatrix}\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}},
+\]
+
+\end_inset
+
+es un método de sumabilidad permanente.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Toeplitsz:
+\series default
+ 
+\begin_inset Formula $A\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+ es un método de sumabilidad permanente si y sólo si 
+\begin_inset Formula $\sup_{n}\sum_{m}|A_{nm}|<\infty$
+\end_inset
+
+, 
+\begin_inset Formula $\forall m\in\mathbb{N},\lim_{n}A_{nm}=0$
+\end_inset
+
+ y 
+\begin_inset Formula $\lim_{n}\sum_{m}A_{nm}=1$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Convergencia puntual de series de Fourier de funciones continuas
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $X\coloneqq\{f\in{\cal C}([-\pi,\pi])\mid f(\pi)=f(-\pi)\}$
+\end_inset
+
+; para 
+\begin_inset Formula $k\in\mathbb{Z}$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in L^{2}([-\pi,\pi])$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\hat{f}(k)\coloneqq\sum_{k=-n}^{n}\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)\text{e}^{-\text{i}kt}\dif t
+\]
+
+\end_inset
+
+el 
+\begin_inset Formula $k$
+\end_inset
+
+-ésimo coeficiente de Fourier de 
+\begin_inset Formula $f$
+\end_inset
+
+ y, para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $s_{n}:L^{2}([-\pi,\pi])\to\mathbb{R}$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+s_{n}(f)(x)\coloneqq\sum_{k=-n}^{n}\hat{f}(k)\text{e}^{\text{i}kx},
+\]
+
+\end_inset
+
+entonces:
+\end_layout
+
+\begin_layout Enumerate
+Como 
+\series bold
+teorema
+\series default
+, existe 
+\begin_inset Formula $F$
+\end_inset
+
+ 
+\begin_inset Formula $G_{\delta}$
+\end_inset
+
+ denso en 
+\begin_inset Formula $X$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $f\in F$
+\end_inset
+
+, 
+\begin_inset Formula $\{x\in[-\pi,\pi]\mid\sup_{n}|s_{n}(f)(x)|\}$
+\end_inset
+
+ es 
+\begin_inset Formula $G_{\delta}$
+\end_inset
+
+ no numerable y denso en 
+\begin_inset Formula $[-\pi,\pi]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $f\in X$
+\end_inset
+
+ de clase 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in[-\pi,\pi]$
+\end_inset
+
+, 
+\begin_inset Formula $\lim_{n}s_{n}(f)(x)=f(x)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Para todo 
+\begin_inset Formula $f\in L^{2}([-\pi,\pi])$
+\end_inset
+
+ y casi todo 
+\begin_inset Formula $x\in[-\pi,\pi]$
+\end_inset
+
+, 
+\begin_inset Formula $\lim_{n}s_{n}(f)(x)=f(x)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Teorema de la aplicación abierta
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio normado, 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es 
+\series bold
+CS-compacto
+\series default
+ si para 
+\begin_inset Formula $\{x_{n}\}_{n}\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\lambda_{n}\}_{n}\subseteq[0,1]$
+\end_inset
+
+ con 
+\begin_inset Formula $\sum_{n}\lambda_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{n}\lambda_{n}x_{n}$
+\end_inset
+
+ converge a un punto de 
+\begin_inset Formula $A$
+\end_inset
+
+, y es 
+\series bold
+CS-cerrado
+\series default
+ si para 
+\begin_inset Formula $\{x_{n}\}_{n}\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\lambda_{n}\}_{n}\subseteq[0,1]$
+\end_inset
+
+ con 
+\begin_inset Formula $\sum_{n}\lambda_{n}=1$
+\end_inset
+
+, si 
+\begin_inset Formula $\sum_{n}\lambda_{n}x_{n}$
+\end_inset
+
+ converge, lo hace un punto de 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio normado:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de Banach, 
+\begin_inset Formula $B_{X}$
+\end_inset
+
+ es CS-compacta.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Todo cerrado convexo es CS-cerrado.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Todo CS-compacto es CS-cerrado y acotado, y el recíproco se cumple si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de Banach.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es CS-cerrado, 
+\begin_inset Formula $\mathring{A}=\mathring{\overline{A}}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios normados y 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+, si 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es CS-compacto, 
+\begin_inset Formula $T(A)$
+\end_inset
+
+ también.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de la aplicación abierta:
+\series default
+ Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Banach, 
+\begin_inset Formula $Y$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio normado y 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+, si 
+\begin_inset Formula $\text{Im}T$
+\end_inset
+
+ es de segunda categoría en 
+\begin_inset Formula $Y$
+\end_inset
+
+, 
+\begin_inset Formula $T$
+\end_inset
+
+ es suprayectiva y abierta e 
+\begin_inset Formula $Y$
+\end_inset
+
+ es un espacio de Banach.
+ 
+\series bold
+Demostración:
+\series default
+ Como 
+\begin_inset Formula $B_{X}$
+\end_inset
+
+ es CS-compacto, 
+\begin_inset Formula $T(B_{X})$
+\end_inset
+
+ también y por tanto es CS-cerrado, y si fuera raro, como el producto por
+ un 
+\begin_inset Formula $n>0$
+\end_inset
+
+ es un homeomorfismo, 
+\begin_inset Formula $nT(B_{X})$
+\end_inset
+
+ sería raro y 
+\begin_inset Formula $T(X)=T(\bigcup_{n\in\mathbb{N}^{*}}nB_{X})=\bigcup_{n\in\mathbb{N}^{*}}nT(B_{X})$
+\end_inset
+
+ sería de primera categoría
+\begin_inset Formula $\#$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\mathring{\overbrace{T(B_{X})}}=\mathring{\overline{T(B_{X})}}\neq\emptyset$
+\end_inset
+
+ y existen 
+\begin_inset Formula $y_{0}\in Y$
+\end_inset
+
+ y 
+\begin_inset Formula $r>0$
+\end_inset
+
+ con 
+\begin_inset Formula $B(y_{0},r)\subseteq T(B_{X})$
+\end_inset
+
+, pero una bola cerrada en el origen es simétrica y 
+\begin_inset Formula $T$
+\end_inset
+
+ conserva simetrías, luego 
+\begin_inset Formula $B(-y_{0},r)\subseteq T(B_{X})$
+\end_inset
+
+ y 
+\begin_inset Formula $B(0,r)\subseteq\frac{1}{2}B_{Y}(-y_{0},r)+\frac{1}{2}B_{Y}(y_{0},r)\subseteq\frac{1}{2}T(B_{X})+\frac{1}{2}T(B_{X})\subseteq T(B_{X})$
+\end_inset
+
+.
+ Así, si 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es abierto, para 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ existe 
+\begin_inset Formula $\delta>0$
+\end_inset
+
+ con 
+\begin_inset Formula $\overline{B(x,\delta)}=x+\delta B_{X}\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $B(Tx,\delta r)=Tx+\delta B(0,r)\subseteq Tx+\delta T(B_{X})=T(x+\delta B_{X})\subseteq T(A)$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $T$
+\end_inset
+
+ es abierta, y para 
+\begin_inset Formula $y\in Y$
+\end_inset
+
+, 
+\begin_inset Formula $y\in B(0,2\Vert y\Vert)=\frac{2\Vert y\Vert}{r}B(0,r)\subseteq T(\frac{2}{r}\Vert y\Vert B_{X})\subseteq T(X)$
+\end_inset
+
+ y 
+\begin_inset Formula $T$
+\end_inset
+
+ es suprayectiva.
+ Finalmente, sea 
+\begin_inset Formula $\{y_{n}\}_{n}\subseteq Y$
+\end_inset
+
+ con 
+\begin_inset Formula $\sum_{n}\Vert y_{n}\Vert<\infty$
+\end_inset
+
+, existe 
+\begin_inset Formula $\{x_{n}\}_{n}\subseteq X$
+\end_inset
+
+ con cada 
+\begin_inset Formula $Tx_{n}=y_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert x_{n}\Vert\leq\frac{2}{r}\Vert y_{n}\Vert$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\sum_{n}\Vert x_{n}\Vert<\infty$
+\end_inset
+
+ y, por ser 
+\begin_inset Formula $X$
+\end_inset
+
+ completo, existe 
+\begin_inset Formula $x'\coloneqq\sum_{n}x_{n}$
+\end_inset
+
+, y por la continuidad de 
+\begin_inset Formula $T$
+\end_inset
+
+, 
+\begin_inset Formula $Tx'=\sum_{n}y_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Entonces, si 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ son de Banach, 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+ es suprayectiva si y sólo si es abierta.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Para esto último hace falta que 
+\begin_inset Formula $Y$
+\end_inset
+
+ sea completo; la identidad 
+\begin_inset Formula $I\in{\cal L}({\cal C}^{1}([0,1]),|\cdot|),({\cal C}^{1}([0,1]),\Vert\cdot\Vert_{\infty}))$
+\end_inset
+
+ con 
+\begin_inset Formula $|x|\coloneqq\Vert x\Vert_{\infty}+\Vert x'\Vert_{\infty}$
+\end_inset
+
+, el dominio es completo e 
+\begin_inset Formula $I$
+\end_inset
+
+ es suprayectiva pero no abierta.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+También hace falta que 
+\begin_inset Formula $X$
+\end_inset
+
+ sea completo; si 
+\begin_inset Formula $(e_{i})_{i\in I}$
+\end_inset
+
+ es una base algebraica no numerable de 
+\begin_inset Formula $\ell^{p}$
+\end_inset
+
+ y 
+\begin_inset Formula $X$
+\end_inset
+
+ es 
+\begin_inset Formula $\ell^{p}$
+\end_inset
+
+ con la norma 
+\begin_inset Formula $\left|\sum_{i}a_{i}e_{i}\right|\coloneqq\sum_{i}|a_{i}|$
+\end_inset
+
+, donde la suma es finita, la identidad 
+\begin_inset Formula $I\in{\cal L}(X,\ell^{p})$
+\end_inset
+
+ es suprayectiva pero no abierta.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema del homomorfismo de Banach:
+\series default
+ Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ un espacio de Banach e 
+\begin_inset Formula $Y$
+\end_inset
+
+ un espacio normado, 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+ es un 
+\series bold
+homomorfismo topológico
+\series default
+ si la restricción a la imagen 
+\begin_inset Formula $T:X\to\text{Im}T$
+\end_inset
+
+ es abierta, si y sólo si 
+\begin_inset Formula $\text{Im}T$
+\end_inset
+
+ es completo.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{TS}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal T}'$
+\end_inset
+
+ son 
+\series bold
+comparables
+\series default
+ si 
+\begin_inset Formula ${\cal T}\subseteq{\cal T}'$
+\end_inset
+
+ o 
+\begin_inset Formula ${\cal T}'\subseteq{\cal T}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ son espacios de Banach y 
+\begin_inset Formula $T:X\to Y$
+\end_inset
+
+ es un isomorfismo algebraico continuo o abierto, 
+\begin_inset Formula $T$
+\end_inset
+
+ es un isomorfismo topológico.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Dos normas completas en 
+\begin_inset Formula $X$
+\end_inset
+
+ que definen topologías comparables son equivalentes.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si un espacio de Banach 
+\begin_inset Formula $X$
+\end_inset
+
+ es suma directa interna 
+\begin_inset Formula $M\oplus N$
+\end_inset
+
+ con 
+\begin_inset Formula $M$
+\end_inset
+
+ y 
+\begin_inset Formula $N$
+\end_inset
+
+ cerrados, entonces 
+\begin_inset Formula $X$
+\end_inset
+
+ es suma directa topológica de 
+\begin_inset Formula $M$
+\end_inset
+
+ y 
+\begin_inset Formula $N$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Técnica de perturbaciones
+\end_layout
+
+\begin_layout Standard
+El problema de Cauchy
+\begin_inset Formula 
+\[
+\left\{ \begin{array}{rl}
+a_{n}(t)x^{(n)}(t)+\dots+a_{1}(t)\dot{x}(t)+a_{0}x(t) & =y(t),\\
+x(a),\dot{x}(a),\dots,x^{(n-1)}(a) & =0
+\end{array}\right.
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $a_{i},y\in{\cal C}([a,b])$
+\end_inset
+
+ tiene solución única 
+\begin_inset Formula $x\in{\cal C}^{(n)}([a,b])$
+\end_inset
+
+ y sus soluciones dependen continuamente del término independiente.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Teorema de la gráfica cerrada
+\end_layout
+
+\begin_layout Standard
+Una función 
+\begin_inset Formula $f:X\to Y$
+\end_inset
+
+ entre espacios topológicos Hausdorff tiene 
+\series bold
+gráfica cerrada
+\series default
+ si 
+\begin_inset Formula $\text{Graf}f\coloneqq\{(x,f(x))\}_{x\in X}$
+\end_inset
+
+ es cerrado en 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua, tiene gráfica cerrada.
+ El recíproco no es cierto; si 
+\begin_inset Formula $X$
+\end_inset
+
+ tiene dos topologías Hausdorff 
+\begin_inset Formula ${\cal T}\prec{\cal S}$
+\end_inset
+
+, 
+\begin_inset Formula $1_{X}:(X,{\cal T})\to(X,{\cal S})$
+\end_inset
+
+ no es continua pero tiene gráfica cerrada.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de la gráfica cerrada:
+\series default
+ Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios de Banach, 
+\begin_inset Formula $T:X\to Y$
+\end_inset
+
+ lineal es continua si y sólo si tiene gráfica cerrada.
+ 
+\series bold
+Demostración:
+\series default
+ Como 
+\begin_inset Formula $x\mapsto(x,Tx)$
+\end_inset
+
+ es lineal, 
+\begin_inset Formula $\text{Graf}T$
+\end_inset
+
+ es un espacio vectorial, las proyecciones canónicas 
+\begin_inset Formula $P_{1}:\text{Graf}T\to X$
+\end_inset
+
+ y 
+\begin_inset Formula $P_{2}:\text{Graf}T\to Y$
+\end_inset
+
+ son lineales y continuas en 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ con la topología producto generada por 
+\begin_inset Formula $\Vert\cdot\Vert_{1}$
+\end_inset
+
+, y como 
+\begin_inset Formula $P_{1}$
+\end_inset
+
+ es biyectiva y por tanto un isomorfismo algebraico, si 
+\begin_inset Formula $\text{Graf}T$
+\end_inset
+
+ es cerrada, es completa al serlo 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ y 
+\begin_inset Formula $P_{1}$
+\end_inset
+
+ es un isomorfismo topológico, con lo que 
+\begin_inset Formula $T=P_{2}\circ P_{1}^{-1}$
+\end_inset
+
+ es continua.
+\end_layout
+
+\begin_layout Standard
+Aquí hace falta que 
+\begin_inset Formula $X$
+\end_inset
+
+ sea completo; la derivada 
+\begin_inset Formula $T:({\cal C}^{1}([0,1]),\Vert\cdot\Vert_{\infty})\to({\cal C}([0,1]),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ es lineal con gráfica cerrada pero no continua.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+También hace falta que 
+\begin_inset Formula $Y$
+\end_inset
+
+ sea completo; si 
+\begin_inset Formula $(e_{i})_{i\in I}$
+\end_inset
+
+ es una base algebraica no numerable de 
+\begin_inset Formula $\ell^{p}$
+\end_inset
+
+ con cada 
+\begin_inset Formula $\Vert e_{i}\Vert=1$
+\end_inset
+
+ y 
+\begin_inset Formula $X$
+\end_inset
+
+ es 
+\begin_inset Formula $\ell^{p}$
+\end_inset
+
+ con la norma 
+\begin_inset Formula $\left|\sum_{i}a_{i}e_{i}\right|\coloneqq\sum_{i}|a_{i}|$
+\end_inset
+
+ siendo la suma finita, la identidad 
+\begin_inset Formula $\ell^{p}\to X$
+\end_inset
+
+ tiene gráfica cerrada pero no es continua.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Separación de puntos
+\end_layout
+
+\begin_layout Standard
+Un conjunto de funciones 
+\begin_inset Formula $F\subseteq B^{A}$
+\end_inset
+
+ 
+\series bold
+separa
+\series default
+ los puntos de 
+\begin_inset Formula $A$
+\end_inset
+
+ si 
+\begin_inset Formula $\forall x,y\in A,(x\neq y\implies\exists f\in F:f(x)\neq f(y))$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de Banach con las normas 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert\cdot\Vert'$
+\end_inset
+
+ y 
+\begin_inset Formula $F\subseteq(X,\Vert\cdot\Vert)^{*}\cap(X,\Vert\cdot\Vert')^{*}$
+\end_inset
+
+ separa los puntos de 
+\begin_inset Formula $X$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert\cdot\Vert'$
+\end_inset
+
+ son equivalentes, y en particular 
+\begin_inset Formula $(X,\Vert\cdot\Vert)^{*}=(X,\Vert\cdot\Vert')^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dos normas completas en el mismo espacio vectorial producen el mismo dual
+ topológico si y sólo si son equivalentes.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Bases de Schauder
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+base de Schauder
+\series default
+ en un espacio normado 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es una sucesión 
+\begin_inset Formula $\{x_{n}\}_{n}\subseteq S_{X}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in X,\exists!\{\lambda_{n}\}_{n}\subseteq\mathbb{K}:x=\sum_{n}\lambda_{n}x_{n}$
+\end_inset
+
+.
+ La sucesión 
+\begin_inset Formula $(e_{n})_{n}$
+\end_inset
+
+ de vectores que valen 1 en la coordenada 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima y 0 en el resto es base de Schauder de 
+\begin_inset Formula $c_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $\ell^{p}$
+\end_inset
+
+ para 
+\begin_inset Formula $p\in[1,\infty)$
+\end_inset
+
+, y 
+\begin_inset Formula $({\cal C}([0,1]),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ y 
+\begin_inset Formula $(L^{p}([0,1]),\Vert\cdot\Vert_{p})$
+\end_inset
+
+ también admiten bases de Schauder.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Todo espacio normado con base de Schauder es separable, pero el recíproco
+ no se cumple.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de las bases de Schauder de Banach:
+\series default
+ Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach con base de Schauder 
+\begin_inset Formula $(x_{n})_{n}$
+\end_inset
+
+, las 
+\series bold
+funciones coordenada
+\series default
+ 
+\begin_inset Formula $f_{n}:X\to\mathbb{K}$
+\end_inset
+
+ dadas por 
+\begin_inset Formula $f_{n}(\sum_{n}\lambda_{n}x_{n})\coloneqq\lambda_{n}$
+\end_inset
+
+ son continuas, y de hecho existe 
+\begin_inset Formula $M>0$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert f_{n}\Vert\leq M$
+\end_inset
+
+ para cada 
+\begin_inset Formula $n$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Pares duales
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+par dual
+\series default
+ es un par 
+\begin_inset Formula $\langle F,G\rangle$
+\end_inset
+
+ de 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios vectoriales con una función bilineal 
+\begin_inset Formula $\langle\cdot,\cdot\rangle:F\times G\to\mathbb{K}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall y\in G,(\langle\cdot,y\rangle=0\implies y=0)$
+\end_inset
+
+ y 
+\begin_inset Formula $\forall x\in G,(\langle x,\cdot\rangle=0\implies x=0)$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+topología débil de 
+\begin_inset Formula $F$
+\end_inset
+
+ inducida por 
+\begin_inset Formula $G$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $\sigma(F,G)$
+\end_inset
+
+, a la topología más gruesa en 
+\begin_inset Formula $F$
+\end_inset
+
+ para la que las 
+\begin_inset Formula $\{\langle\cdot,y\rangle\}_{y\in G}$
+\end_inset
+
+ son continuas, generada por la familia de seminormas 
+\begin_inset Formula $\{|\langle\cdot,y\rangle|\}_{y\in G}$
+\end_inset
+
+, y 
+\series bold
+topología débil de 
+\begin_inset Formula $G$
+\end_inset
+
+ inducida por 
+\begin_inset Formula $F$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $\sigma(G,F)$
+\end_inset
+
+, a la topología más gruesa en 
+\begin_inset Formula $F$
+\end_inset
+
+ para la que las 
+\begin_inset Formula $\{\langle x,\cdot\rangle\}_{x\in F}$
+\end_inset
+
+ son continuas, generada por la familia de seminormas 
+\begin_inset Formula $\{|\langle f,\cdot\rangle|\}_{x\in F}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un espacio vectorial y 
+\begin_inset Formula $E^{*}$
+\end_inset
+
+ su dual algebraico, 
+\begin_inset Formula $\langle E,E^{*}\rangle$
+\end_inset
+
+ es un par dual con la 
+\series bold
+aplicación bilineal natural
+\series default
+ 
+\begin_inset Formula $\langle x,f\rangle\coloneqq f(x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c., 
+\begin_inset Formula $\langle E,E'\rangle$
+\end_inset
+
+ es un par dual con la aplicación bilineal natural, el 
+\series bold
+par dual canónico
+\series default
+, y llamamos 
+\series bold
+topología débil de 
+\begin_inset Formula $E$
+\end_inset
+
+
+\series default
+ a 
+\begin_inset Formula $\sigma(E,E')$
+\end_inset
+
+ y 
+\series bold
+topología débil* de 
+\begin_inset Formula $E'$
+\end_inset
+
+
+\series default
+ a 
+\begin_inset Formula $\sigma(E',E)$
+\end_inset
+
+, que es Hausdorff e inducida por 
+\begin_inset Formula ${\cal T}_{\text{p}}(E)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $I$
+\end_inset
+
+ es un conjunto, 
+\begin_inset Formula $\langle\mathbb{K}^{I},\mathbb{K}^{(I)}\rangle$
+\end_inset
+
+ es un par dual con 
+\begin_inset Formula $\langle(\lambda_{i})_{i\in I},(\xi_{i})_{i\in I}\rangle=\sum_{i\in I}\lambda_{i}\xi_{i}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K$
+\end_inset
+
+ es compacto, 
+\begin_inset Formula $E\coloneqq(C(K),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ y 
+\begin_inset Formula $F\coloneqq\text{span}\{f\mapsto f(x)\}_{x\in K}\leq(C(K),\Vert\cdot\Vert_{\infty})^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\langle E,F\rangle$
+\end_inset
+
+ es un par dual con la aplicación bilineal natural, y 
+\begin_inset Formula $\sigma(E,F)={\cal T}_{\text{p}}(K)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\langle F,G\rangle$
+\end_inset
+
+ es un par dual, una forma lineal 
+\begin_inset Formula $f:F\to\mathbb{K}$
+\end_inset
+
+ es 
+\begin_inset Formula $\sigma(F,G)$
+\end_inset
+
+-continua si y sólo si existe 
+\begin_inset Formula $y\in G$
+\end_inset
+
+, necesariamente único, con 
+\begin_inset Formula $f=\langle\cdot,y\rangle$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c., 
+\begin_inset Formula $(E,\sigma(E,E'))'=E'$
+\end_inset
+
+ e, identificando 
+\begin_inset Formula $x\in E$
+\end_inset
+
+ con 
+\begin_inset Formula $\hat{x}\in E''$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\hat{x}(f)\coloneqq f(x)$
+\end_inset
+
+, 
+\begin_inset Formula $(E',\sigma(E',E))'=E$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\langle F,G\rangle$
+\end_inset
+
+ es un par dual con función bilineal 
+\begin_inset Formula $\langle\cdot,\cdot\rangle$
+\end_inset
+
+ y 
+\begin_inset Formula $H\leq G$
+\end_inset
+
+, 
+\begin_inset Formula $\langle\cdot,\cdot\rangle$
+\end_inset
+
+ induce un par dual en 
+\begin_inset Formula $\langle F,H\rangle$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $G=\overline{H}$
+\end_inset
+
+ en 
+\begin_inset Formula $\sigma(G,F)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c., 
+\begin_inset Formula $E'$
+\end_inset
+
+ es 
+\begin_inset Formula $\sigma(E^{*},E)$
+\end_inset
+
+-denso en el dual algebraico 
+\begin_inset Formula $E^{*}$
+\end_inset
+
+, con lo que las formas lineales se aproximan por formas lineales continuas.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dado un par dual 
+\begin_inset Formula $\langle F,G\rangle$
+\end_inset
+
+, llamamos 
+\series bold
+polar
+\series default
+ (
+\series bold
+absoluta
+\series default
+) de 
+\begin_inset Formula $A\subseteq F$
+\end_inset
+
+ a 
+\begin_inset Formula $A^{\circ}\coloneqq\{y\in G\mid\sup_{x\in A}|\langle x,y\rangle|\leq1\}$
+\end_inset
+
+ y 
+\series bold
+bipolar
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ a 
+\begin_inset Formula $A^{\circ\circ}\subseteq F$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es un espacio normado, 
+\begin_inset Formula $B_{X}^{\circ}=B_{X^{*}}$
+\end_inset
+
+ y 
+\begin_inset Formula $B_{X}^{\circ\circ}=B_{X}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\langle F,G\rangle$
+\end_inset
+
+ es un par dual y 
+\begin_inset Formula $M\leq F$
+\end_inset
+
+, 
+\begin_inset Formula $M^{\circ}=\{y\in G\mid\langle M,y\rangle=0\}\eqqcolon M^{\bot}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\langle F,G\rangle$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-par dual, 
+\begin_inset Formula $A,B,A_{i}\subseteq F$
+\end_inset
+
+ para 
+\begin_inset Formula $i\in I$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in\mathbb{K}^{*}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A^{\circ}$
+\end_inset
+
+ es absolutamente convexo y cerrado en 
+\begin_inset Formula $\sigma(G,F)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $B\subseteq A\implies A^{\circ}\subseteq B^{\circ}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(\alpha A)^{\circ}=\alpha^{-1}A^{\circ}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A\subseteq A^{\circ\circ}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A^{\circ}\subseteq A^{\circ\circ\circ}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(\bigcup_{i\in I}A_{i})^{\circ}=\bigcap_{i\in I}A_{i}^{\circ}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema del bipolar:
+\series default
+ Si 
+\begin_inset Formula $\langle F,G\rangle$
+\end_inset
+
+ es un par dual y 
+\begin_inset Formula $A\subseteq F$
+\end_inset
+
+, 
+\begin_inset Formula $A^{\circ\circ}=\overline{\Gamma(A)}$
+\end_inset
+
+ en 
+\begin_inset Formula $\sigma(F,G)$
+\end_inset
+
+ (la envoltura absolutamente convexa cerrada).
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c., 
+\begin_inset Formula $M\subseteq E'$
+\end_inset
+
+ es 
+\series bold
+equicontinuo
+\series default
+ si 
+\begin_inset Formula $\forall\varepsilon>0,\exists U\in{\cal E}(0_{E}):\forall f\in M,\forall x\in U,|f(x)|<\varepsilon$
+\end_inset
+
+, y una 
+\series bold
+familia fundamental de equicontinuos
+\series default
+ es un 
+\begin_inset Formula ${\cal E}\subseteq{\cal P}(E')$
+\end_inset
+
+ con los elementos equicontinuos tal que para 
+\begin_inset Formula $M\subseteq E'$
+\end_inset
+
+ equicontinuo existe 
+\begin_inset Formula $N\in{\cal E}$
+\end_inset
+
+ que contiene a 
+\begin_inset Formula $M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $(E,{\cal T})$
+\end_inset
+
+ es un e.l.c.:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $U\in{\cal E}(0)$
+\end_inset
+
+, 
+\begin_inset Formula $U^{\circ}\subseteq E'$
+\end_inset
+
+ es equicontinuo, y si 
+\begin_inset Formula $M\subseteq E'$
+\end_inset
+
+ es equicontinuo, 
+\begin_inset Formula $M^{\circ}\in{\cal E}(0)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+ es base de entornos de 0 en 
+\begin_inset Formula $E$
+\end_inset
+
+, 
+\begin_inset Formula $\{U^{\circ}\}_{U\in{\cal U}}$
+\end_inset
+
+ es una familia fundamental de equicontinuos.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ es una familia fundamental de equicontinuos, 
+\begin_inset Formula $\{M^{\circ}\}_{M\in{\cal E}}$
+\end_inset
+
+ es una base de entornos de 0.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+ es la topología de convergencia uniforme sobre los equicontinuos de 
+\begin_inset Formula $E'$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\langle F,G\rangle$
+\end_inset
+
+ un par dual y 
+\begin_inset Formula ${\cal S}\subseteq{\cal P}(G)$
+\end_inset
+
+ una familia de subconjuntos 
+\begin_inset Formula $\sigma(F,G)$
+\end_inset
+
+-cerrados absolutamente convexos, en 
+\begin_inset Formula $\sigma(F,G)$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\left(\bigcap{\cal S}\right)^{\circ}=\overline{\Gamma\left(\bigcup_{S\in{\cal S}}S^{\circ}\right)}.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Alaoglu-Bourbaki:
+\series default
+ Si 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c., todo equicontinuo 
+\begin_inset Formula $H$
+\end_inset
+
+ de 
+\begin_inset Formula $E'$
+\end_inset
+
+ es relativamente compacto en 
+\begin_inset Formula $\sigma(E',E)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es un espacio normado, 
+\begin_inset Formula $B_{X^{*}}$
+\end_inset
+
+ es compacta en 
+\begin_inset Formula $\sigma(X^{*},X)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Lema de aproximación:
+\series default
+ Sean 
+\begin_inset Formula $E$
+\end_inset
+
+ es un e.l.c., 
+\begin_inset Formula $S\subseteq E$
+\end_inset
+
+ cerrado y absolutamente convexo y 
+\begin_inset Formula $f:E\to\mathbb{K}$
+\end_inset
+
+ lineal, 
+\begin_inset Formula $f|_{S}$
+\end_inset
+
+ es continua si y sólo si 
+\begin_inset Formula $\forall\varepsilon>0,\exists g\in E':\sup_{x\in S}|g(x)-f(x)|<\varepsilon$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de completitud de Grothendieck:
+\series default
+ Sean 
+\begin_inset Formula $E$
+\end_inset
+
+ un e.l.c.
+ y 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ el conjunto de los equicontinuos de 
+\begin_inset Formula $E'$
+\end_inset
+
+, 
+\begin_inset Formula $\hat{E}\coloneqq\{x\in(E')^{*}\mid\forall M\in{\cal E},x|_{M}\text{ continuo en }\sigma(E',E)\}$
+\end_inset
+
+ con la topología de convergencia uniforme sobre 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ es un modelo para la compleción de 
+\begin_inset Formula $E$
+\end_inset
+
+, es decir, 
+\begin_inset Formula $E$
+\end_inset
+
+ es denso en 
+\begin_inset Formula $\hat{E}$
+\end_inset
+
+ y 
+\begin_inset Formula $\hat{E}$
+\end_inset
+
+ es completo.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así:
+\end_layout
+
+\begin_layout Enumerate
+Un e.l.c.
+ 
+\begin_inset Formula $E$
+\end_inset
+
+ es completo si y sólo si toda 
+\begin_inset Formula $y:E'\to\mathbb{K}$
+\end_inset
+
+ lineal 
+\begin_inset Formula $\sigma(E',E)$
+\end_inset
+
+-continua sobre los equicontinuos de 
+\begin_inset Formula $E'$
+\end_inset
+
+ es 
+\begin_inset Formula $\sigma(E',E)$
+\end_inset
+
+-continua en 
+\begin_inset Formula $E'$
+\end_inset
+
+, si y sólo si está en 
+\begin_inset Formula $E$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Un 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ normado es de Banach si y sólo si toda 
+\begin_inset Formula $x:X^{*}\to\mathbb{K}$
+\end_inset
+
+ lineal 
+\begin_inset Formula $\sigma(X^{*},X)$
+\end_inset
+
+-continua en 
+\begin_inset Formula $B_{X^{*}}$
+\end_inset
+
+ es 
+\begin_inset Formula $\sigma(X^{*},X)$
+\end_inset
+
+-continua en 
+\begin_inset Formula $X^{*}$
+\end_inset
+
+, si y sólo si está en 
+\begin_inset Formula $E$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es normado, 
+\begin_inset Formula $K\coloneqq(B_{X^{*}},\sigma(X^{*},X))$
+\end_inset
+
+ e 
+\begin_inset Formula $\iota:X\hookrightarrow C(K)$
+\end_inset
+
+ es la identificación estándar en el bidual:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\iota:(X,\Vert\cdot\Vert)\hookrightarrow(C(K),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ e 
+\begin_inset Formula $\iota:(X,\Vert\cdot\Vert)\hookrightarrow(C(K),{\cal T}_{\text{p}}(K))$
+\end_inset
+
+ son isomorfismos isométricos sobre su imagen.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de Banach, 
+\begin_inset Formula $(X,\sigma(X,X^{*}))$
+\end_inset
+
+ se identifica con un subespacio cerrado de 
+\begin_inset Formula $(C(K),{\cal T}_{\text{p}}(K))$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Espacios reflexivos
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es un espacio normado y 
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+ la topología asociada a 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\sigma(X,X^{*})$
+\end_inset
+
+ es más gruesa que 
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma(X^{*},X)$
+\end_inset
+
+ es más gruesa que la asociada a la norma dual.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\sigma(X,X^{*})$
+\end_inset
+
+ es metrizable si y sólo si 
+\begin_inset Formula $X$
+\end_inset
+
+ es dimensión finita, en cuyo caso es igual a 
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ convexo es cerrado en 
+\begin_inset Formula $\sigma(X,X^{*})$
+\end_inset
+
+ si y sólo si lo es en 
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un espacio de Banach es 
+\series bold
+reflexivo
+\series default
+ si la identificación estándar 
+\begin_inset Formula $\hat{}:X\to X^{**}$
+\end_inset
+
+ es suprayectiva.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p\in(1,\infty)$
+\end_inset
+
+ y 
+\begin_inset Formula $(\Omega,\Sigma,\mu)$
+\end_inset
+
+ es un espacio de medida, 
+\begin_inset Formula $(L^{p}(\mu),\Vert\cdot\Vert_{p})$
+\end_inset
+
+ es reflexivo.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(c_{0},\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ no es reflexivo.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Goldstine:
+\series default
+ Sea 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ normado, 
+\begin_inset Formula $B_{X}$
+\end_inset
+
+ es denso en 
+\begin_inset Formula $(B_{X^{**}},\sigma(X^{**},X^{*}))$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de caracterización de la reflexividad:
+\series default
+ Un espacio de Banach 
+\begin_inset Formula $X$
+\end_inset
+
+ es reflexivo si y sólo si 
+\begin_inset Formula $B_{X}$
+\end_inset
+
+ es compacta en 
+\begin_inset Formula $\sigma(X,X^{*})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X$
+\end_inset
+
+ es separable si y sólo si 
+\begin_inset Formula $(B_{X^{*}},\sigma(X^{*},X))$
+\end_inset
+
+ es metrizable.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X^{*}$
+\end_inset
+
+ es separable si y solo si 
+\begin_inset Formula $(B_{X},\sigma(X,X^{*}))$
+\end_inset
+
+ es metrizable.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $X^{*}$
+\end_inset
+
+es separable, 
+\begin_inset Formula $X$
+\end_inset
+
+ es separable.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio reflexivo:
+\end_layout
+
+\begin_layout Enumerate
+Todo subespacio cerrado es reflexivo.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X$
+\end_inset
+
+ es separable si y sólo si lo es 
+\begin_inset Formula $X^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un espacio de Banach 
+\begin_inset Formula $X$
+\end_inset
+
+ es reflexivo si y sólo si lo es 
+\begin_inset Formula $X^{*}$
+\end_inset
+
+ con la norma dual.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+Todo espacio de dimensión finita es reflexivo.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\ell^{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\ell^{\infty}$
+\end_inset
+
+ son reflexivos.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Ni 
+\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ ni su dual son reflexivos.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Ni 
+\begin_inset Formula $L^{1}([a,b])$
+\end_inset
+
+ ni 
+\begin_inset Formula $L^{\infty}([a,b])$
+\end_inset
+
+ son reflexivos.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un espacio normado 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es 
+\series bold
+uniformemente convexo
+\series default
+ si 
+\begin_inset Formula 
+\[
+\forall\varepsilon>0,\exists\delta>0:\forall x,y\in B_{X},\left(\Vert x-y\Vert\geq\varepsilon\implies\left\Vert \frac{x+y}{2}\right\Vert \leq1-\delta\right),
+\]
+
+\end_inset
+
+si y sólo si 
+\begin_inset Formula 
+\[
+\forall\{x_{n}\}_{n},\{y_{n}\}_{n}\subseteq B_{X},\left(\lim_{n}\left\Vert \frac{x_{n}+y_{n}}{2}\right\Vert =1\implies\lim_{n}\Vert x_{n}-y_{n}\Vert=0\right),
+\]
+
+\end_inset
+
+en cuyo caso 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+ es 
+\series bold
+uniformemente convexa
+\series default
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Toda norma uniformemente convexa es estrictamente convexa.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Todo espacio prehilbertiano es uniformemente convexo.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+En un espacio normado 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+, 
+\begin_inset Formula $f:X\to\mathbb{R}$
+\end_inset
+
+ es 
+\series bold
+uniformemente diferenciable Fréchet
+\series default
+ en 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ si existe 
+\begin_inset Formula $\lim_{t\to0}\sup_{h\in B_{X}}\frac{f(x+th)-f(x)}{t}$
+\end_inset
+
+.
+
+\series bold
+ Primer teorema de Šmulian:
+\series default
+ Un espacio de Banach 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es uniformemente convexo si y sólo si para 
+\begin_inset Formula $f\in B_{X^{*}}$
+\end_inset
+
+, la norma dual es uniformemente diferenciable Fréchet en todo 
+\begin_inset Formula $B_{X^{*}}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Milman:
+\series default
+ Todo espacio de Banach con norma uniformemente convexa es reflexivo.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach y 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, un 
+\series bold
+
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+-árbol diádico
+\series default
+ con 
+\series bold
+raíz
+\series default
+ 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ de longitud 
+\begin_inset Formula $N\in\mathbb{N}\cup\{\infty\}$
+\end_inset
+
+ es una familia 
+\begin_inset Formula $\{x_{s}\}_{s\in\bigcup_{i=0}^{n}\{\pm1\}^{n}}\subseteq X$
+\end_inset
+
+ tal que 
+\begin_inset Formula $x_{\emptyset}=x$
+\end_inset
+
+ y, para 
+\begin_inset Formula $s\in\bigcup_{i=0}^{n-1}\{\pm1\}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $x_{s}=\frac{x_{s(-1)}+x_{s1}}{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert x_{s(-1)}-x_{s1}\Vert\geq\varepsilon$
+\end_inset
+
+.
+ Un espacio de Banach 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es 
+\series bold
+superreflexivo
+\series default
+ si para 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ existe 
+\begin_inset Formula $N\in\mathbb{N}$
+\end_inset
+
+ tal que todo 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+-árbol diádico contenido en 
+\begin_inset Formula $B_{X}$
+\end_inset
+
+ tiene longitud máxima 
+\begin_inset Formula $N$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $X$
+\end_inset
+
+ admite una norma uniformemente convexa equivalente a 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $p\in(1,\infty)$
+\end_inset
+
+ y 
+\begin_inset Formula $(\Omega,\Sigma,\mu)$
+\end_inset
+
+ es un espacio de medida, 
+\begin_inset Formula $L^{p}(\Omega,\Sigma,\mu)$
+\end_inset
+
+ es uniformemente convexo y reflexivo, y si 
+\begin_inset Formula $q\in(1,\infty)$
+\end_inset
+
+ es tal que 
+\begin_inset Formula $\frac{1}{p}+\frac{1}{q}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\Phi:L^{q}(\mu)\to L^{p}(\mu)^{*}$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+\Phi(g)(f)\coloneqq\int_{\Omega}fg\dif\mu
+\]
+
+\end_inset
+
+ es un isomorfismo isométrico.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Propiedad de Schur:
+\series default
+ En 
+\begin_inset Formula $\ell^{1}$
+\end_inset
+
+, las sucesiones convergentes en la topología asociada a la norma y en 
+\begin_inset Formula $\sigma(\ell^{1},\ell^{\infty})$
+\end_inset
+
+ son las mismas, pese a que son topologías distintas.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Segundo teorema de Šmulian:
+\series default
+ Si 
+\begin_inset Formula $(X,\Vert\cdot\Vert)$
+\end_inset
+
+ es normado, un subespacio de 
+\begin_inset Formula $(X,\sigma(X,X^{*}))$
+\end_inset
+
+ es compacto si y sólo si es compacto por sucesiones.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un subconjunto de 
+\begin_inset Formula $(\ell^{1},\Vert\cdot\Vert_{1})$
+\end_inset
+
+ es débilmente compacto (compacto con la topología débil) si y sólo si es
+ compacto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document